A Beginner's Guide To Structural Equation Ing Beginners 3rd Ed

User Manual: Pdf

Open the PDF directly: View PDF .
Page Count: 510

Download
Open PDF In Browser	View PDF

A Beginner’s Guide to

Structural
Equation
Modeling
Third Edition

Randall E. Schumacker
The University of Alabama

Richard G. Lomax
The Ohio State University

Y102005.indb 3

4/3/10 4:25:16 PM

Routledge
Taylor & Francis Group
711 Third Avenue
New York, NY 10017

Routledge
Taylor & Francis Group
27 Church Road
Hove, East Sussex BN3 2FA

International Standard Book Number: 978-1-84169-890-8 (Hardback) 978-1-84169-891-5 (Paperback)
For permission to photocopy or use material electronically from this work, please access www.
copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc.
(CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been
granted a photocopy license by the CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and
are used only for identification and explanation without intent to infringe.
Library of Congress Cataloging-in-Publication Data
Schumacker, Randall E.
A beginner’s guide to structural equation modeling / authors, Randall E.
Schumacker, Richard G. Lomax.-- 3rd ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-84169-890-8 (hardcover : alk. paper) -- ISBN 978-1-84169-891-5
(pbk. : alk. paper)
1. Structural equation modeling. 2. Social sciences--Statistical methods. I.
Lomax, Richard G. II. Title.
QA278.S36 2010
519.5’3--dc22

2010009456

Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the Psychology Press Web site at
http://www.psypress.com

Y102005.indb 4

4/3/10 4:25:16 PM

Contents
About the Authors............................................................................................xv
Preface.............................................................................................................. xvii
1 Introduction................................................................................................. 1
1.1 What Is Structural Equation Modeling?........................................ 2
1.2 History of Structural Equation Modeling..................................... 4
1.3 Why Conduct Structural Equation Modeling?............................. 6
1.4 Structural Equation Modeling Software Programs..................... 8
1.5 Summary.......................................................................................... 10
References................................................................................................... 11
2 Data Entry and Data Editing Issues...................................................... 13
2.1 Data Entry........................................................................................ 14
2.2 Data Editing Issues......................................................................... 18
2.2.1 Measurement Scale............................................................ 18
2.2.2 Restriction of Range.......................................................... 19
2.2.3 Missing Data....................................................................... 20
2.2.4 LISREL–PRELIS Missing Data Example........................ 21
2.2.5 Outliers................................................................................ 27
2.2.6 Linearity.............................................................................. 27
2.2.7 Nonnormality..................................................................... 28
2.3 Summary.......................................................................................... 29
References................................................................................................... 31
3 Correlation................................................................................................. 33
3.1 Types of Correlation Coefficients.................................................. 33
3.2 Factors Affecting Correlation Coefficients.................................. 35
3.2.1 Level of Measurement and Range of Values.................. 35
3.2.2 Nonlinearity....................................................................... 36
3.2.3 Missing Data....................................................................... 38
3.2.4 Outliers................................................................................ 39
3.2.5 Correction for Attenuation............................................... 39
3.2.6 Nonpositive Definite Matrices......................................... 40
3.2.7 Sample Size......................................................................... 41
3.3 Bivariate, Part, and Partial Correlations...................................... 42
3.4 Correlation versus Covariance...................................................... 46
3.5 Variable Metrics (Standardized versus Unstandardized)......... 47
3.6 Causation Assumptions and Limitations.................................... 48
3.7 Summary.......................................................................................... 49
References................................................................................................... 51
vii

Y102005.indb 7

4/3/10 4:25:17 PM

viii

Contents

4 SEM Basics................................................................................................. 55
4.1 Model Specification......................................................................... 55
4.2 Model Identification........................................................................ 56
4.3 Model Estimation............................................................................ 59
4.4 Model Testing.................................................................................. 63
4.5 Model Modification........................................................................ 64
4.6 Summary.......................................................................................... 67
References................................................................................................... 69
5 Model Fit..................................................................................................... 73
5.1 Types of Model-Fit Criteria............................................................ 74
5.1.1 LISREL–SIMPLIS Example............................................... 77
5.1.1.1 Data...................................................................... 77
5.1.1.2 Program............................................................... 80
5.1.1.3 Output.................................................................. 81
5.2 Model Fit........................................................................................... 85
5.2.1 Chi-Square (χ 2)................................................................... 85
5.2.2 Goodness-of-Fit Index (GFI) and Adjusted
Goodness-of-Fit Index (AGFI).......................................... 86
5.2.3 Root-Mean-Square Residual Index (RMR)..................... 87
5.3 Model Comparison......................................................................... 88
5.3.1 Tucker–Lewis Index (TLI)................................................. 88
5.3.2 Normed Fit Index (NFI) and Comparative Fit
Index (CFI).......................................................................... 88
5.4 Model Parsimony............................................................................ 89
5.4.1 Parsimony Normed Fit Index (PNFI).............................. 90
5.4.2 Akaike Information Criterion (AIC)............................... 90
5.4.3 Summary............................................................................. 91
5.5 Parameter Fit.................................................................................... 92
5.6 Power and Sample Size.................................................................. 93
5.6.1 Model Fit............................................................................. 94
5.6.1.1 Power.................................................................... 94
5.6.1.2 Sample Size......................................................... 99
5.6.2 Model Comparison.......................................................... 108
5.6.3 Parameter Significance.....................................................111
5.6.4 Summary............................................................................113
5.7 Two-Step Versus Four-Step Approach to Modeling.................114
5.8 Summary.........................................................................................116
Chapter Footnote......................................................................................118
Standard Errors.........................................................................................118
Chi-Squares................................................................................................118
References................................................................................................. 120

Y102005.indb 8

4/3/10 4:25:17 PM

Contents

6 Regression Models................................................................................. 125
6.1 Overview........................................................................................ 126
6.2 An Example.................................................................................... 130
6.3 Model Specification....................................................................... 130
6.4 Model Identification...................................................................... 131
6.5 Model Estimation.......................................................................... 131
6.6 Model Testing................................................................................ 133
6.7 Model Modification...................................................................... 134
6.8 Summary........................................................................................ 135
6.8.1 Measurement Error.......................................................... 136
6.8.2 Additive Equation............................................................ 137
Chapter Footnote..................................................................................... 138
Regression Model with Intercept Term...................................... 138
LISREL–SIMPLIS Program (Intercept Term)....................................... 138
References................................................................................................. 139
7 Path Models............................................................................................. 143
7.1 An Example.................................................................................... 144
7.2 Model Specification....................................................................... 147
7.3 Model Identification...................................................................... 150
7.4 Model Estimation.......................................................................... 151
7.5 Model Testing................................................................................ 154
7.6 Model Modification...................................................................... 155
7.7 Summary........................................................................................ 156
Appendix: LISREL–SIMPLIS Path Model Program............................ 156
Chapter Footnote..................................................................................... 158
Another Traditional Non-SEM Path Model-Fit Index............. 158
LISREL–SIMPLIS program.......................................................... 158
References..................................................................................................161
8 Confirmatory Factor Models................................................................ 163
8.1 An Example.................................................................................... 164
8.2 Model Specification....................................................................... 166
8.3 Model Identification.......................................................................167
8.4 Model Estimation.......................................................................... 169
8.5 Model Testing................................................................................ 170
8.6 Model Modification...................................................................... 173
8.7 Summary.........................................................................................174
Appendix: LISREL–SIMPLIS Confirmatory Factor Model Program.....174
References................................................................................................. 177
9 Developing Structural Equation Models: Part I.............................. 179
9.1 Observed Variables and Latent Variables.................................. 180
9.2 Measurement Model..................................................................... 184

Y102005.indb 9

4/3/10 4:25:17 PM

Contents

9.3 Structural Model........................................................................... 186
9.4 Variances and Covariance Terms............................................... 189
9.5 Two-Step/Four-Step Approach................................................... 191
9.6 Summary........................................................................................ 192
References................................................................................................. 193
10 Developing Structural Equation Models: Part II............................. 195
10.1 An Example.................................................................................... 195
10.2 Model Specification....................................................................... 197
10.3 Model Identification...................................................................... 200
10.4 Model Estimation.......................................................................... 202
10.5 Model Testing................................................................................ 203
10.6 Model Modification...................................................................... 205
10.7 Summary........................................................................................ 207
Appendix: LISREL–SIMPLIS Structural Equation Model Program...... 207
References................................................................................................. 208
11 Reporting SEM Research: Guidelines and Recommendations.... 209
11.1 Data Preparation........................................................................... 212
11.2 Model Specification....................................................................... 213
11.3 Model Identification...................................................................... 215
11.4 Model Estimation...........................................................................216
11.5 Model Testing................................................................................ 217
11.6 Model Modification...................................................................... 218
11.7 Summary........................................................................................ 219
References................................................................................................. 220
12 Model Validation.................................................................................... 223
Key Concepts............................................................................................ 223
12.1 Multiple Samples........................................................................... 223
12.1.1 Model A Computer Output............................................ 226
12.1.2 Model B Computer Output............................................. 227
12.1.3 Model C Computer Output............................................ 228
12.1.4 Model D Computer Output............................................ 229
12.1.5 Summary........................................................................... 229
12.2 Cross Validation............................................................................ 229
12.2.1 ECVI................................................................................... 230
12.2.2 CVI..................................................................................... 231
12.3 Bootstrap........................................................................................ 234
12.3.1 PRELIS Graphical User Interface................................... 234
12.3.2 LISREL and PRELIS Program Syntax........................... 237
12.4 Summary........................................................................................ 241
References................................................................................................. 243

Y102005.indb 10

4/3/10 4:25:17 PM

Contents

13 Multiple Sample, Multiple Group, and Structured
Means Models......................................................................................... 245
13.1 Multiple Sample Models.............................................................. 245
Sample 1......................................................................................... 247
Sample 2......................................................................................... 247
13.2 Multiple Group Models................................................................ 250
13.2.1 Separate Group Models................................................... 251
13.2.2 Similar Group Model...................................................... 255
13.2.3 Chi-Square Difference Test............................................. 258
13.3 Structured Means Models........................................................... 259
13.3.1 Model Specification and Identification......................... 259
13.3.2 Model Fit........................................................................... 261
13.3.3 Model Estimation and Testing....................................... 261
13.4 Summary........................................................................................ 263
Suggested Readings................................................................................. 267
Multiple Samples.......................................................................... 267
Multiple Group Models............................................................... 267
Structured Means Models............................................................ 267
Chapter Footnote..................................................................................... 268
SPSS................................................................................................. 268
References................................................................................................. 269
14 Second-Order, Dynamic, and Multitrait Multimethod Models......271
14.1 Second-Order Factor Model........................................................ 271
14.1.1 Model Specification and Identification......................... 271
14.1.2 Model Estimation and Testing....................................... 272
14.2 Dynamic Factor Model..................................................................274
14.3 Multitrait Multimethod Model (MTMM).................................. 277
14.3.1 Model Specification and Identification......................... 279
14.3.2 Model Estimation and Testing....................................... 280
14.3.3 Correlated Uniqueness Model....................................... 281
14.4 Summary........................................................................................ 286
Suggested Readings................................................................................. 290
Second-Order Factor Models....................................................... 290
Dynamic Factor Models............................................................... 290
Multitrait Multimethod Models.................................................. 290
Correlated Uniqueness Model.................................................... 291
References................................................................................................. 291
15 Multiple Indicator–Multiple Indicator Cause, Mixture,
and Multilevel Models.......................................................................... 293
15.1 Multiple Indicator–Multiple Cause (MIMIC) Models.............. 293
15.1.1 Model Specification and Identification......................... 294
15.1.2 Model Estimation and Model Testing........................... 294

Y102005.indb 11

4/3/10 4:25:17 PM

xii

Contents

15.1.3 Model Modification......................................................... 297
Goodness-of-Fit Statistics............................................... 297
Measurement Equations................................................. 297
Structural Equations........................................................ 298
15.2 Mixture Models............................................................................. 298
15.2.1 Model Specification and Identification......................... 299
15.2.2 Model Estimation and Testing....................................... 301
15.2.3 Model Modification......................................................... 302
15.2.4 Robust Statistic................................................................. 305
15.3 Multilevel Models......................................................................... 307
15.3.1 Constant Effects............................................................... 313
15.3.2 Time Effects...................................................................... 313
15.3.3 Gender Effects.................................................................. 315
15.3.4 Multilevel Model Interpretation.....................................318
15.3.5 Intraclass Correlation...................................................... 319
15.3.6 Deviance Statistic............................................................. 320
15.4 Summary........................................................................................ 320
Suggested Readings................................................................................. 324
Multiple Indicator–Multiple Cause Models.............................. 324
Mixture Models............................................................................. 325
Multilevel Models......................................................................... 325
References................................................................................................. 325
16 Interaction, Latent Growth, and Monte Carlo Methods................. 327
16.1 Interaction Models........................................................................ 327
16.1.1 Categorical Variable Approach...................................... 328
16.1.2 Latent Variable Interaction Model................................. 331
16.1.2.1 Computing Latent Variable Scores................ 331
16.1.2.2 Computing Latent Interaction Variable........ 333
16.1.2.3 Interaction Model Output............................... 335
16.1.2.4 Model Modification.......................................... 336
16.1.2.5 Structural Equations—No Latent
Interaction Variable.......................................... 336
16.1.3 Two-Stage Least Squares (TSLS) Approach................. 337
16.2 Latent Growth Curve Models..................................................... 341
16.2.1 Latent Growth Curve Program...................................... 343
16.2.2 Model Modification......................................................... 344
16.3 Monte Carlo Methods................................................................... 345
16.3.1 PRELIS Simulation of Population Data........................ 346
16.3.2 Population Data from Specified
Covariance Matrix........................................................... 352
16.3.2.1 SPSS Approach................................................. 352
16.3.2.2 SAS Approach................................................... 354
16.3.2.3 LISREL Approach............................................. 355

Y102005.indb 12

4/3/10 4:25:18 PM

Contents

xiii

16.3.3 Covariance Matrix from Specified Model.................... 359
16.4 Summary........................................................................................ 365
Suggested Readings................................................................................. 368
Interaction Models........................................................................ 368
Latent Growth-Curve Models..................................................... 368
Monte Carlo Methods................................................................... 368
References................................................................................................. 369
17 Matrix Approach to Structural Equation Modeling........................ 373
17.1 General Overview of Matrix Notation....................................... 373
17.2 Free, Fixed, and Constrained Parameters.................................. 379
17.3 LISREL Model Example in Matrix Notation............................. 382
LISREL8 Matrix Program Output (Edited and Condensed).. 385
17.4 Other Models in Matrix Notation............................................... 400
17.4.1 Path Model........................................................................ 400
17.4.2 Multiple-Sample Model.................................................. 404
17.4.3 Structured Means Model................................................ 405
17.4.4 Interaction Models........................................................... 410
PRELIS Computer Output........................................................... 412
LISREL Interaction Computer Output........................................416
17.5 Summary........................................................................................ 421
References................................................................................................. 423
Appendix A: Introduction to Matrix Operations.................................... 425
Appendix B: Statistical Tables.................................................................... 439
Answers to Selected Exercises.................................................................... 449
Author Index................................................................................................... 489
Subject Index.................................................................................................. 495

Y102005.indb 13

4/3/10 4:25:18 PM

About the Authors
RANDALL E. SCHUMACKER received his Ph.D. in educational psychology from Southern Illinois University. He is currently professor of educational research at the University of Alabama, where he teaches courses
in structural equation modeling, multivariate statistics, multiple regression, and program evaluation. His research interests are varied, including
modeling interaction in SEM, robust statistics (normal scores, centering,
and variance inflation factor issues), and SEM specification search issues
as well as measurement model issues related to estimation, mixed-item
formats, and reliability.
He has published in several journals including Academic Medicine,
Educational and Psychological Measurement, Journal of Applied Measurement,
Journal of Educational and Behavioral Statistics, Journal of Research Methodology,
Multiple Linear Regression Viewpoints, and Structural Equation Modeling.
He has served on the editorial boards of numerous journals and is a
member of the American Educational Research Association, American
Psychological Association—Division 5, as well as past-president of the
Southwest Educational Research Association, and emeritus editor of
Structural Equation Modeling journal. He can be contacted at the University
of Alabama College of Education.
RICHARD G. LOMAX received his Ph.D. in educational research methodology from the University of Pittsburgh. He is currently a professor in
the School of Educational Policy and Leadership, Ohio State University,
where he teaches courses in structural equation modeling, statistics, and
quantitative research methodology.
His research primarily focuses on models of literacy acquisition, multivariate statistics, and assessment. He has published in such diverse journals as Parenting: Science and Practice, Understanding Statistics: Statistical
Issues in Psychology, Education, and the Social Sciences, Violence Against
Women, Journal of Early Adolescence, and Journal of Negro Education. He has
served on the editorial boards of numerous journals, and is a member of
the American Educational Research Association, the American Statistical
Association, and the National Reading Conference. He can be contacted at
Ohio State University College of Education and Human Ecology.

Y102005.indb 15

4/3/10 4:25:18 PM

Preface

Approach
This book presents a basic introduction to structural equation modeling
(SEM). Readers will find that we have kept to our tradition of keeping
examples rudimentary and easy to follow. The reader is provided with
a review of correlation and covariance, followed by multiple regression,
path, and factor analyses in order to better understand the building blocks
of SEM. The book describes a basic structural equation model followed by
the presentation of several different types of structural equation models.
Our approach in the text is both conceptual and application oriented.
Each chapter covers basic concepts, principles, and practice and then
utilizes SEM software to provide meaningful examples. Each chapter also
features an outline, key concepts, a summary, numerous examples from
a variety of disciplines, tables, and figures, including path diagrams, to
assist with conceptual understanding. Chapters with examples follow the
conceptual sequence of SEM steps known as model specification, identification, estimation, testing, and modification.
The book now uses LISREL 8.8 student version to make the software and
examples readily available to readers. Please be aware that the student
version, although free, does not contain all of the functional features as a
full licensed version. Given the advances in SEM software over the past
decade, you should expect updates and patches of this software package
and therefore become familiar with any new features as well as explore the
excellent library of examples and help materials. The LISREL 8.8 student
version is an easy-to-use Windows PC based program with pull-down
menus, dialog boxes, and drawing tools. To access the program, and/or
if you’re a Mac user and are interested in learning about Mac availability,
please check with Scientific Software (http://www.ssicentral.com). There
is also a hotlink to the Scientific Software site from the book page for A
Beginner’s Guide to Structural Equation Modeling, 3rd edition on the Textbook
Resources tab at www.psypress.com.
The SEM model examples in the book do not require complicated programming skills nor does the reader need an advanced understanding of
statistics and matrix algebra to understand the model applications. We have
provided a chapter on the matrix approach to SEM as well as an appendix
on matrix operations for the interested reader. We encourage the understanding of the matrices used in SEM models, especially for some of the
more advanced SEM models you will encounter in the research literature.
xvii

Y102005.indb 17

4/3/10 4:25:18 PM

xviii

Preface

Goals and Content Coverage
Our main goal in this third edition is for students and researchers to be
able to conduct their own SEM model analyses, as well as be able to understand and critique published SEM research. These goals are supported by
the conceptual and applied examples contained in the book and several
journal article references for each advanced SEM model type. We have
also included a SEM checklist to guide your model analysis according to
the basic steps a researcher takes.
As for content coverage, the book begins with an introduction to SEM
(what it is, some history, why conduct it, and what software is available),
followed by chapters on data entry and editing issues, and correlation.
These early chapters are critical to understanding how missing data, nonnormality, scale of measurement, non-linearity, outliers, and restriction of
range in scores affects SEM analysis. Chapter 4 lays out the basic steps of
model specification, identification, estimation, testing, and modification,
followed by Chapter 5, which covers issues related to model fit indices,
power and sample size. Chapters 6 through 10 follow the basic SEM steps
of modeling, with actual examples from different disciplines, using regression, path, confirmatory factor and structural equation models. Logically
the next chapter presents information about reporting SEM research and
includes a SEM checklist to guide decision-making. Chapter 12 presents
different approaches to model validation, an important final step after
obtaining an acceptable theoretical model. Chapters 13 through 16 provide
SEM examples that introduce many of the different types of SEM model
applications. The final chapter describes the matrix approach to structural
equation modeling by using examples from the previous chapters.
Theoretical models are present in every discipline, and therefore can be
formulated and tested. This third edition expands SEM models and applications to provide the students and researchers in medicine, political science, sociology, education, psychology, business, and the biological sciences
the basic concepts, principles, and practice necessary to test their theoretical models. We hope you become more familiar with structural equation
modeling after reading the book, and use SEM in your own research.

New to the Third Edition
The first edition of this book was one of the first books published on SEM,
while the second edition greatly expanded knowledge of advanced SEM
models. Since that time, we have had considerable experience utilizing the

Y102005.indb 18

4/3/10 4:25:18 PM

Preface

xix

book in class with our students. As a result of those experiences, the third
edition represents a more useable book for teaching SEM. As such it is an
ideal text for introductory graduate level courses in structural equation
modeling or factor analysis taught in departments of psychology, education, business, and other social and healthcare sciences. An understanding of correlation is assumed.
The third edition offers several new surprises, namely:
1. Our instruction and examples are now based on freely available
software: LISREL 8.8 student version.
2. More examples presented from more disciplines, including input,
output, and screenshots.
3. Every chapter has been updated and enhanced with additional
material.
4. A website with raw data sets for the book’s examples and exercises so they can be used with any SEM program, all of the book’s
exercises, hotlinks to related websites, and answers to all of the
exercises for instructors only. To access the website visit the book
page or the Textbook Resource page at www.psypress.com.
5. Expanded coverage of advanced models with more on multiplegroup, multi-level, and mixture modeling (Chs. 13 and 15), secondorder and dynamic factor models (Ch. 14), and Monte Carlo
methods (Ch. 16).
6. Increased coverage of sample size and power (Ch. 5), including
software programs, and reporting research (Ch. 11).
7. New journal article references help readers better understand
published research (Chs. 13–17).
8. Troubleshooting tips on how to address the most frequently
encountered problems are found in Chapters 3 and 11.
9. Chapters 13 to 16 now include additional SEM model examples.
10. 25% new exercises with answers to half in the back of the book
for student review (and answers to all for instructors only on the
book and/or Textbook Resource page at www.psypress.com).
11. Added Matrix examples for several models in Chapter 17.
12. Updated references in all chapters on all key topics.
Overall, we believe this third edition is a more complete book that can
be used to teach a full course in SEM. The past several years have seen an
explosion in SEM coursework, books, websites, and training courses. We
are proud to have been considered a starting point for many beginner’s
to SEM. We hope you find that this third edition expands on many of the
programming tools, trends and topics in SEM today.

Y102005.indb 19

4/3/10 4:25:18 PM

Preface

Acknowledgments
The third edition of this book represents more than thirty years of interacting with our colleagues and students who use structural equation
modeling. As before, we are most grateful to the pioneers in the field of
structural equation modeling, particularly to Karl Jöreskog, Dag Sörbom,
Peter Bentler, James Arbuckle, and Linda and Bengt Muthèn. These individuals have developed and shaped the new advances in the SEM field as
well as the content of this book, plus provided SEM researchers with software programs. We are also grateful to Gerhard Mels who answered our
questions and inquiries about SEM programming problems in the chapters. We also wish to thank the reviewers: James Leeper, The University
of Alabama, Philip Smith, Augusta State University, Phil Wood, the
University of Missouri–Columbia, and Ke-Haie Yuan, the University of
Notre Dame.
This book was made possible through the encouragement of Debra
Riegert at Routledge/Taylor & Francis who insisted it was time for a third
edition. We wish to thank her and her editorial assistant, Erin M. Flaherty,
for coordinating all of the activity required to get a book into print. We
also want to thank Suzanne Lassandro at Taylor & Francis Group, LLC
for helping us through the difficult process of revisions, galleys, and final
book copy.
Randall E. Schumacker
The University of Alabama
Richard G. Lomax
The Ohio State University

Y102005.indb 20

4/3/10 4:25:18 PM

1
Introduction

Key Concepts
Latent and observed variables
Independent and dependent variables
Types of models
Regression
Path
Confirmatory factor
Structural equation
History of structural equation modeling
Structural equation modeling software programs
Structural equation modeling can be easily understood if the researcher
has a grounding in basic statistics, correlation, and regression analysis.
The first three chapters provide a brief introduction to structural equation
modeling (SEM), basic data entry, and editing issues in statistics, and concepts related to the use of correlation coefficients in structural equation
modeling. Chapter 4 covers the essential concepts of SEM: model specification, identification, estimation, testing, and modification. This basic
understanding provides the framework for understanding the material
presented in chapters 5 through 8 on model-fit indices, regression analysis, path analysis, and confirmatory factor analysis models (measurement
models), which form the basis for understanding the structural equation
models (latent variable models) presented in chapters 9 and 10. Chapter 11
provides guidance on reporting structural equation modeling research.
Chapter 12 addresses techniques used to establish model validity and
generalization of findings. Chapters 13 to 16 present many of the advanced
SEM models currently appearing in journal articles: multiple group, multiple indicators–multiple causes, mixture, multilevel, structured means,
multitrait–multimethod, second-order factor, dynamic factor, interaction
1

Y102005.indb 1

3/22/10 3:24:44 PM

A Beginner’s Guide to Structural Equation Modeling

models, latent growth curve models, and Monte Carlo studies. Chapter 17
presents matrix notation for one of our SEM applications, covers the different matrices used in structural equation modeling, and presents multiple
regression and path analysis solutions using matrix algebra. We include
an introduction to matrix operations in the Appendix for readers who
want a more mathematical understanding of matrix operations. To start
our journey of understanding, we first ask, What is structural equation
modeling? Then, we give a brief history of SEM, discuss the importance of
SEM, and note the availability of SEM software programs.

1.1 What Is Structural Equation Modeling?
Structural equation modeling (SEM) uses various types of models to
depict relationships among observed variables, with the same basic goal
of providing a quantitative test of a theoretical model hypothesized by
the researcher. More specifically, various theoretical models can be tested
in SEM that hypothesize how sets of variables define constructs and
how these constructs are related to each other. For example, an educational researcher might hypothesize that a student’s home environment
influences her later achievement in school. A marketing researcher may
hypothesize that consumer trust in a corporation leads to increased product sales for that corporation. A health care professional might believe
that a good diet and regular exercise reduce the risk of a heart attack.
In each example, the researcher believes, based on theory and empirical
research, sets of variables define the constructs that are hypothesized to be
related in a certain way. The goal of SEM analysis is to determine the extent to
which the theoretical model is supported by sample data. If the sample data
support the theoretical model, then more complex theoretical models can be
hypothesized. If the sample data do not support the theoretical model, then
either the original model can be modified and tested, or other theoretical
models need to be developed and tested. Consequently, SEM tests theoretical models using the scientific method of hypothesis testing to advance our
understanding of the complex relationships among constructs.
SEM can test various types of theoretical models. Basic models include
regression (chapter 6), path (chapter 7), and confirmatory factor (chapter 8) models. Our reason for covering these basic models is that they
provide a basis for understanding structural equation models (chapters
9 and 10). To better understand these basic models, we need to define a
few terms. First, there are two major types of variables: latent variables
and observed variables. Latent variables (constructs or factors) are variables that are not directly observable or measured. Latent variables are

Y102005.indb 2

3/22/10 3:24:44 PM

Introduction

indirectly observed or measured, and hence are inferred from a set of
observed variables that we actually measure using tests, surveys, and
so on. For example, intelligence is a latent variable that represents a psychological construct. The confidence of consumers in American business
is another latent variable, one representing an economic construct. The
physical condition of adults is a third latent variable, one representing a
health-related construct.
The observed, measured, or indicator variables are a set of variables that
we use to define or infer the latent variable or construct. For example, the
Wechsler Intelligence Scale for Children—Revised (WISC-R) is an instrument that produces a measured variable (scores), which one uses to infer
the construct of a child’s intelligence. Additional indicator variables, that
is, intelligence tests, could be used to indicate or define the construct of
intelligence (latent variable). The Dow-Jones index is a standard measure
of the American corporate economy construct. Other measured variables
might include gross national product, retail sales, or export sales. Blood
pressure is one of many health-related variables that could indicate a
latent variable defined as “fitness.” Each of these observed or indicator
variables represent one definition of the latent variable. Researchers use
sets of indicator variables to define a latent variable; thus, other measurement instruments are used to obtain indicator variables, for example, the
Stanford–Binet Intelligence Scale, the NASDAQ index, and an individual’s
cholesterol level, respectively.
Variables, whether they are observed or latent, can also be defined
as either independent variables or dependent variables. An independent
variable is a variable that is not influenced by any other variable in
the model. A dependent variable is a variable that is influenced by
another variable in the model. Let us return to the previous examples
and specify the independent and dependent variables. The educational
researcher hypothesizes that a student’s home environment (independent latent variable) influences school achievement (dependent latent
variable). The marketing researcher believes that consumer trust in a
corporation (independent latent variable) leads to increased product
sales (dependent latent variable). The health care professional wants to
determine whether a good diet and regular exercise (two independent
latent variables) influence the frequency of heart attacks (dependent
latent variable).
The basic SEM models in chapters 6 through 8 illustrate the use of
observed variables and latent variables when defined as independent
or dependent. A regression model consists solely of observed variables
where a single dependent observed variable is predicted or explained by
one or more independent observed variables; for example, a parent’s education level (independent observed variable) is used to predict his or her
child’s achievement score (dependent observed variable). A path model is

Y102005.indb 3

3/22/10 3:24:44 PM

A Beginner’s Guide to Structural Equation Modeling

also specified entirely with observed variables, but the flexibility allows
for multiple independent observed variables and multiple dependent
observed variables—for example, export sales, gross national product,
and NASDAQ index influence consumer trust and consumer spending
(dependent observed variables). Path models, therefore, test more complex models than regression models. Confirmatory factor models consist of observed variables that are hypothesized to measure one or more
latent variables (independent or dependent); for example, diet, exercise,
and physiology are observed measures of the independent latent variable
“fitness.” An understanding of these basic models will help in understanding structural equation modeling, which combines path and factor
analytic models. Structural equation models consist of observed variables
and latent variables, whether independent or dependent; for example, an
independent latent variable (home environment) influences a dependent
latent variable (achievement), where both types of latent variables are
measured, defined, or inferred by multiple observed or measured indicator variables.

1.2 History of Structural Equation Modeling
To discuss the history of structural equation modeling, we explain the following four types of related models and their chronological order of development: regression, path, confirmatory factor, and structural equation
models.
The first model involves linear regression models that use a correlation
coefficient and the least squares criterion to compute regression weights.
Regression models were made possible because Karl Pearson created a
formula for the correlation coefficient in 1896 that provides an index for
the relationship between two variables (Pearson, 1938). The regression
model permits the prediction of dependent observed variable scores
(Y scores), given a linear weighting of a set of independent observed
scores (X scores) that minimizes the sum of squared residual error values. The mathematical basis for the linear regression model is found in
basic algebra. Regression analysis provides a test of a theoretical model
that may be useful for prediction (e.g., admission to graduate school or
budget projections). In an example study, regression analysis was used
to predict student exam scores in statistics (dependent variable) from a
series of collaborative learning group assignments (independent variables; Delucchi, 2006). The results provided some support for collaborative learning groups improving statistics exam performance, although
not for all tasks.

Y102005.indb 4

3/22/10 3:24:44 PM

Introduction

Some years later, Charles Spearman (1904, 1927) used the correlation
coefficient to determine which items correlated or went together to create
the factor model. His basic idea was that if a set of items correlated or
went together, individual responses to the set of items could be summed
to yield a score that would measure, define, or infer a construct. Spearman
was the first to use the term factor analysis in defining a two-factor construct for a theory of intelligence. D.N. Lawley and L.L. Thurstone in 1940
further developed applications of factor models, and proposed instruments (sets of items) that yielded observed scores from which constructs
could be inferred. Most of the aptitude, achievement, and diagnostic
tests, surveys, and inventories in use today were created using factor analytic techniques. The term confirmatory factor analysis (CFA) is used today
based in part on earlier work by Howe (1955), Anderson and Rubin (1956),
and Lawley (1958). The CFA method was more fully developed by Karl
Jöreskog in the 1960s to test whether a set of items defined a construct.
Jöreskog completed his dissertation in 1963, published the first article on
CFA in 1969, and subsequently helped develop the first CFA software program. Factor analysis has been used for over 100 years to create measurement instruments in many academic disciplines, while today CFA is used
to test the existence of these theoretical constructs. In an example study,
CFA was used to confirm the “Big Five” model of personality by Goldberg
(1990). The five-factor model of extraversion, agreeableness, conscientiousness, neuroticism, and intellect was confirmed through the use of multiple
indicator variables for each of the five hypothesized factors.
Sewell Wright (1918, 1921, 1934), a biologist, developed the third type of
model, a path model. Path models use correlation coefficients and regression analysis to model more complex relationships among observed
variables. The first applications of path analysis dealt with models of
animal behavior. Unfortunately, path analysis was largely overlooked
until econometricians reconsidered it in the 1950s as a form of simultaneous equation modeling (e.g., H. Wold) and sociologists rediscovered it in
the 1960s (e.g., O. D. Duncan and H. M. Blalock). In many respects, path
analysis involves solving a set of simultaneous regression equations that
theoretically establish the relationship among the observed variables in
the path model. In an example path analysis study, Walberg’s theoretical
model of educational productivity was tested for fifth- through eighthgrade students (Parkerson et al., 1984). The relations among the following variables were analyzed in a single model: home environment, peer
group, media, ability, social environment, time on task, motivation, and
instructional strategies. All of the hypothesized paths among those variables were shown to be statistically significant, providing support for the
educational productivity model.
The final model type is structural equation modeling (SEM). SEM models essentially combine path models and confirmatory factor models;

Y102005.indb 5

3/22/10 3:24:44 PM

A Beginner’s Guide to Structural Equation Modeling

that is, SEM models incorporate both latent and observed variables. The
early development of SEM models was due to Karl Jöreskog (1969, 1973),
Ward Keesling (1972), and David Wiley (1973); this approach was initially
known as the JKW model, but became known as the linear structural relations model (LISREL) with the development of the first software program,
LISREL, in 1973. Since then, many SEM articles have been published; for
example, Shumow and Lomax (2002) tested a theoretical model of parental efficacy for adolescent students. For the overall sample, neighborhood
quality predicted parental efficacy, which predicted parental involvement
and monitoring, both of which predicted academic and social-emotional
adjustment.
Jöreskog and van Thillo originally developed the LISREL software program at the Educational Testing Service (ETS) using a matrix command
language (i.e., involving Greek and matrix notation), which is described
in chapter 17. The first publicly available version, LISREL III, was released
in 1976. Later in 1993, LISREL8 was released; it introduced the SIMPLIS
(SIMPle LISrel) command language in which equations are written
using variable names. In 1999, the first interactive version of LISREL was
released. LISREL8 introduced the dialog box interface using pull-down
menus and point-and-click features to develop models, and the path diagram mode, a drawing program to develop models. Karl Jöreskog was recognized by Cudeck, DuToit, and Sörbom (2001) who edited a Festschrift
in honor of his contributions to the field of structural equation modeling.
Their volume contains chapters by scholars who address the many topics, concerns, and applications in the field of structural equation modeling today, including milestones in factor analysis; measurement models;
robustness, reliability, and fit assessment; repeated measurement designs;
ordinal data; and interaction models. We cover many of these topics in
this book, although not in as great a depth. The field of structural equation modeling across all disciplines has expanded since 1994. Hershberger
(2003) found that between 1994 and 2001 the number of journal articles
concerned with SEM increased, the number of journals publishing SEM
research increased, SEM became a popular choice amongst multivariate
methods, and the journal Structural Equation Modeling became the primary
source for technical developments in structural equation modeling.

1.3 Why Conduct Structural Equation Modeling?
Why is structural equation modeling popular? There are at least four
major reasons for the popularity of SEM. The first reason suggests that
researchers are becoming more aware of the need to use multiple observed

Y102005.indb 6

3/22/10 3:24:45 PM

Introduction

variables to better understand their area of scientific inquiry. Basic statistical methods only utilize a limited number of variables, which are not
capable of dealing with the sophisticated theories being developed. The
use of a small number of variables to understand complex phenomena is
limiting. For instance, the use of simple bivariate correlations is not sufficient for examining a sophisticated theoretical model. In contrast, structural equation modeling permits complex phenomena to be statistically
modeled and tested. SEM techniques are therefore becoming the preferred
method for confirming (or disconfirming) theoretical models in a quantitative fashion.
A second reason involves the greater recognition given to the validity and reliability of observed scores from measurement instruments.
Specifically, measurement error has become a major issue in many disciplines, but measurement error and statistical analysis of data have
been treated separately. Structural equation modeling techniques explicitly take measurement error into account when statistically analyzing
data. As noted in subsequent chapters, SEM analysis includes latent and
observed variables as well as measurement error terms in certain SEM
models.
A third reason pertains to how structural equation modeling has matured
over the past 30 years, especially the ability to analyze more advanced theoretical SEM models. For example, group differences in theoretical models
can be assessed through multiple-group SEM models. In addition, analyzing educational data collected at more than one level—for example, school
districts, schools, and teachers with student data—is now possible using
multilevel SEM modeling. As a final example, interaction terms can now
be included in an SEM model so that main effects and interaction effects
can be tested. These advanced SEM models and techniques have provided
many researchers with an increased capability to analyze sophisticated
theoretical models of complex phenomena, thus requiring less reliance on
basic statistical methods.
Finally, SEM software programs have become increasingly userfriendly. For example, until 1993 LISREL users had to input the program syntax for their models using Greek and matrix notation. At
that time, many researchers sought help because of the complex programming requirement and knowledge of the SEM syntax that was
required. Today, most SEM software programs are Windows-based
and use pull-down menus or drawing programs to generate the program syntax internally. Therefore, the SEM software programs are now
easier to use and contain features similar to other Windows-based
software packages. However, such ease of use necessitates statistical training in SEM modeling and software via courses, workshops,
or textbooks to avoid mistakes and errors in analyzing sophisticated
theoretical models.

Y102005.indb 7

3/22/10 3:24:45 PM

A Beginner’s Guide to Structural Equation Modeling

1.4 Structural Equation Modeling Software Programs
Although the LISREL program was the first SEM software program,
other software programs have subsequently been developed since the
mid-1980s. Some of the other programs include AMOS, EQS, Mx, Mplus,
Ramona, and Sepath, to name a few. These software programs are each
unique in their own way, with some offering specialized features for
conducting different SEM applications. Many of these SEM software
programs provide statistical analysis of raw data (e.g., means, correlations, missing data conventions), provide routines for handling missing
data and detecting outliers, generate the program’s syntax, diagram the
model, and provide for import and export of data and figures of a theoretical model. Also, many of the programs come with sets of data and
program examples that are clearly explained in their user guides. Many
of these software programs have been reviewed in the journal Structural
Equation Modeling.
The pricing information for SEM software varies depending on individual, group, or site license arrangements; corporate versus educational settings; and even whether one is a student or faculty member.
Furthermore, newer versions and updates necessitate changes in pricing. Most programs will run in the Windows environment; some run
on MacIntosh personal computers. We are often asked to recommend
a software package to a beginning SEM researcher; however, given the
different individual needs of researchers and the multitude of different
features available in these programs, we are not able to make such a recommendation. Ultimately the decision depends upon the researcher’s
needs and preferences. Consequently, with so many software packages,
we felt it important to narrow our examples in the book to LISREL–
SIMPLIS programs.
We will therefore be using the LISREL 8.8 student version in the book
to demonstrate the many different SEM applications, including regression models, path models, confirmatory factor models, and the various
SEM models in chapters 13 through 16. The free student version of the
LISREL software program (Windows, Mac, and Linux editions) can be
downloaded from the website: http://www.ssicentral.com/lisrel/student.
html. (Note: The LISREL 8.8 Student Examples folder is placed in the main
directory C:/ of your computer, not the LISREL folder under C:/Program
Files when installing the software.)

Y102005.indb 8

3/22/10 3:24:45 PM

Introduction

Once the LISREL software is downloaded, place an icon on your desktop by creating a shortcut to the LISREL icon. The LISREL icon should
look something like this:

LISREL 8.80 Student.lnk

When you click on the icon, an empty dialog box will appear that should
look like this:

N ote: Nothing appears until you open a program file or data set using
the File or open folder icon; more about this in the next chapter.

We do want to mention the very useful HELP menu. Click on the question mark (?), a HELP menu will appear, then enter Output Questions in
the search window to find answers to key questions you may have when
going over examples in the Third Edition.

Y102005.indb 9

3/22/10 3:25:10 PM

A Beginner’s Guide to Structural Equation Modeling

1.5 Summary
In this chapter we introduced structural equation modeling by describing basic types of variables—that is, latent, observed, independent, and
dependent—and basic types of SEM models—that is, regression, path,
confirmatory factor, and structural equation models. In addition, a brief
history of structural equation modeling was provided, followed by a discussion of the importance of SEM. This chapter concluded with a brief
listing of the different structural equation modeling software programs
and where to obtain the LISREL 8.8 student version for use with examples

Y102005.indb 10

3/22/10 3:25:11 PM

Introduction

in the book, including what the dialog box will first appear like and a very
useful HELP menu.
In chapter 2 we consider the importance of examining data for issues
related to measurement level (nominal, ordinal, interval, or ratio), restriction of range (fewer than 15 categories), missing data, outliers (extreme
values), linearity or nonlinearity, and normality or nonnormality, all of
which can affect statistical methods, and especially SEM applications.
Exercises
1. Define the following terms:
a. Latent variable
b. Observed variable
c. Dependent variable
d. Independent variable
2. Explain the difference between a dependent latent variable and
a dependent observed variable.
3. Explain the difference between an independent latent variable
and an independent observed variable.
4. List the reasons why a researcher would conduct structural
equation modeling.
5. Download and activate the student version of LISREL: http://
www.ssicentral.com
6. Open and import SPSS or data file.

References
Anderson, T. W., & Rubin, H. (1956). Statistical inference in factor analysis. In
J. Neyman (Ed.), Proceedings of the third Berkeley symposium on mathematical statistics and probability, Vol. V (pp. 111–150). Berkeley: University of
California Press.
Cudeck, R., Du Toit, S., & Sörbom, D. (2001) (Eds). Structural equation modeling:
Present and future. A Festschrift in honor of Karl Jöreskog. Lincolnwood, IL:
Scientific Software International.
Delucchi, M. (2006). The efficacy of collaborative learning groups in an undergraduate statistics course. College Teaching, 54, 244–248.
Goldberg, L. (1990). An alternative “description of personality”: Big Five factor
structure. Journal of Personality and Social Psychology, 59, 1216–1229.
Hershberger, S. L. (2003). The growth of structural equation modeling: 1994–2001.
Structural Equation Modeling, 10(1), 35–46.
Howe, W. G. (1955). Some contributions to factor analysis (Report No. ORNL-1919).
Oak Ridge National Laboratory, Oak Ridge, Tennessee.
Jöreskog, K. G. (1963). Statistical estimation in factor analysis: A new technique and its
foundation. Stockholm: Almqvist & Wiksell.

Y102005.indb 11

3/22/10 3:25:11 PM

A Beginner’s Guide to Structural Equation Modeling

Jöreskog, K. G. (1969). A general approach to confirmatory maximum likelihood
factor analysis. Psychometrika, 34, 183–202.
Jöreskog, K. G. (1973). A general method for estimating a linear structural equation
system. In A. S. Goldberger & O. D. Duncan (Eds.), Structural equation models
in the social sciences (pp. 85–112). New York: Seminar.
Keesling, J. W. (1972). Maximum likelihood approaches to causal flow analysis.
Unpublished doctoral dissertation. Chicago: University of Chicago.
Lawley, D. N. (1958). Estimation in factor analysis under various initial assumptions. British Journal of Statistical Psychology, 11, 1–12.
Parkerson, J. A., Lomax, R. G., Schiller, D. P., & Walberg, H. J. (1984). Exploring
causal models of educational achievement. Journal of Educational Psychology,
76, 638–646.
Pearson, E. S. (1938). Karl Pearson. An appreciation of some aspects of his life and work.
Cambridge: Cambridge University Press.
Shumow, L., & Lomax, R. G. (2002). Parental efficacy: Predictor of parenting behavior and adolescent outcomes. Parenting: Science and Practice, 2, 127–150.
Spearman, C. (1904). The proof and measurement of association between two
things. American Journal of Psychology, 15, 72–101.
Spearman, C. (1927). The abilities of man. New York: Macmillan.
Wiley, D. E. (1973). The identification problem for structural equation models with
unmeasured variables. In A. S. Goldberger & O. D. Duncan (Eds.), Structural
equation models in the social sciences (pp. 69–83). New York: Seminar.
Wright, S. (1918). On the nature of size factors. Genetics, 3, 367–374.
Wright, S. (1921). Correlation and causation. Journal of Agricultural Research, 20,
557–585.
Wright, S. (1934). The method of path coefficients. Annals of Mathematical Statistics,
5, 161–215.

Y102005.indb 12

3/22/10 3:25:11 PM

2
Data Entry and Data Editing Issues

Key Concepts
Importing data file
System file
Measurement scale
Restriction of range
Missing data
Outliers
Linearity
Nonnormality
An important first step in using LISREL is to be able to enter raw data and/
or import data, such as files from other programs (SPSS, SAS, EXCEL, etc.).
Other important steps involve being able to use LISREL–PRELIS to save
a system file, as well as output and save files that contain the variance–
covariance matrix, the correlation matrix, means, and standard deviations
of variables so they can be input into command syntax programs. The
LISREL–PRELIS program will be briefly explained in this chapter to demonstrate how it handles raw data entry, importing of data, and the output
of saved files.
There are several key issues in the field of statistics that impact our analyses once data have been imported into a software program. These data
issues are commonly referred to as the measurement scale of variables,
restriction in the range of data, missing data values, outliers, linearity, and
nonnormality. Each of these data issues will be discussed because they
not only affect traditional statistics, but present additional problems and
concerns in structural equation modeling.
We use LISREL software throughout the book, so you will need to use
that software and become familiar with their Web site. You should have
downloaded by now the free student version of the LISREL software.
13

Y102005.indb 13

3/22/10 3:25:11 PM

A Beginner’s Guide to Structural Equation Modeling

We use some of the data and model examples available in the free student version to illustrate SEM applications. (Note: The LISREL 8.8 Student
Examples folder is placed in the main directory C:/ of your computer.)
The free student version of the software has a user guide, help functions,
and tutorials. The Web site also contains important research, documentation, and information about structural equation modeling. However, be
aware that the free student version of the software does not contain the
full capabilities available in their full licensed version (e.g., restricted to
15 observed variables in SEM analyses). These limitations are spelled out
on their Web site.

2.1 Data Entry
The LISREL software program interfaces with PRELIS, a preprocessor of
data prior to running LISREL (matrix command language) or SIMPLIS
(easier-to-use variable name syntax) programs. The newer Interactive
LISREL uses a spreadsheet format for data with pull-down menu options.
LISREL offers several different options for inputting data and importing
files from numerous other programs. The New, Open, and Import Data
functions provide maximum flexibility for inputting data.

The New option permits the creation of a command syntax language
program (PRELIS, LISREL, or SIMPLIS) to read in a PRELIS data file, or

Y102005.indb 14

3/22/10 3:25:12 PM

Data Entry and Data Editing Issues

to open SIMPLIS and LISREL saved projects as well as a previously saved
Path Diagram.

The Open option permits you to browse and locate previously saved
PRELIS (.pr2), LISREL (.ls8), or SIMPLIS (.spl) programs; each with their
unique file extension. The student version has distinct folders containing
several program examples, for example LISREL (LS8EX folder), PRELIS
(PR2EX folder), and SIMPLIS (SPLEX folder).

The Import Data option permits inputting raw data files or SPSS
saved files. The raw data file, lsat6.dat, is in the PRELIS folder (PR2EX).
When selecting this file, you will need to know the number of variables
in the file.

Y102005.indb 15

3/22/10 3:25:12 PM

A Beginner’s Guide to Structural Equation Modeling

An SPSS saved file, data100.sav, is in the SPSS folder (SPSSEX). Once you
open this file, a PRELIS system file is created.

Y102005.indb 16

3/22/10 3:25:13 PM

Data Entry and Data Editing Issues

Once the PRELIS system file becomes active, then it needs to be saved for
future use. (Note: # symbol may appear if columns are to narrow; simply use
your mouse to expand the columns so that the missing values—999999.00
will appear. Also, if you right-mouse click on the variable names, a menu
appears to define missing values, etc.). The PRELIS system file (.psf) activates a pull-down menu that permits data editing features, data transformations, statistical analysis of data, graphical display of data, multilevel
modeling, and many other related features.

Y102005.indb 17

3/22/10 3:25:16 PM

A Beginner’s Guide to Structural Equation Modeling

The statistical analysis of data includes factor analysis, probit regression, least squares regression, and two-stage least squares methods.
Other important data editing features include imputing missing values,
a homogeneity test, creation of normal scores, bootstrapping, and data
output options. The data output options permit saving different types of
variance–covariance matrices and descriptive statistics in files for use in
LISREL and SIMPLIS command syntax programs. This capability is very
important, especially when advanced SEM models are analyzed in chapters 13 to 16. We will demonstrate the use of this Output Options dialog
box in this chapter and in some of our other chapter examples.

2.2 Data Editing Issues
2.2.1 Measurement Scale
How variables are measured or scaled influences the type of statistical
analyses we perform (Anderson, 1961; Stevens, 1946). Properties of scale
also guide our understanding of permissible mathematical operations.

Y102005.indb 18

3/22/10 3:25:17 PM

Data Entry and Data Editing Issues

For example, a nominal variable implies mutually exclusive groups; a
biological gender has two mutually exclusive groups, male and female.
An individual can only be in one of the groups that define the levels
of the variable. In addition, it would not be meaningful to calculate a
mean and a standard deviation on the variable gender. Consequently,
the number or percentage of individuals at each level of the gender
variable is the only mathematical property of scale that makes sense.
An ordinal variable, for example, attitude toward school, that is scaled
strongly agree, agree, neutral, disagree, and strongly disagree, implies mutually exclusive categories that are ordered or ranked. When levels of a
variable have properties of scale that involve mutually exclusive groups
that are ordered, only certain mathematical operations are meaningful, for example, a comparison of ranks between groups. SEM final
exam scores, an example of an interval variable, possesses the property
of scale, implying equal intervals between the data points, but no true
zero point. This property of scale permits the mathematical operation
of computing a mean and a standard deviation. Similarly, a ratio variable, for example, weight, has the property of scale that implies equal
intervals and a true zero point (weightlessness). Therefore, ratio variables also permit mathematical operations of computing a mean and
a standard deviation. Our use of different variables requires us to be
aware of their properties of scale and what mathematical operations
are possible and meaningful, especially in SEM, where variance–
covariance (correlation) matrices are used with means and standard
deviations of variables. Different correlations among variables are
therefore possible depending upon the level of measurement, but they
create unique problems in SEM (see chapter 3). PRELIS designates continuous variables (CO), ordinal variables (OR), and categorical variables (CL) to make these distinctions.
2.2.2 Restriction of Range
Data values at the interval or ratio level of measurement can be further
defined as being discrete or continuous. For example, SEM final exam
scores could be reported in whole numbers (discrete). Similarly, the number of children in a family would be considered a discrete level of measurement—or example, 5 children. In contrast, a continuous variable is
reported using decimal values; for example, a student’s grade point average would be reported as 3.75 on a 5-point scale.
Karl Jöreskog (1996) provided a criterion in the PRELIS program based
on his research that defines whether a variable is ordinal or interval,
based on the presence of 15 distinct scale points. If a variable has fewer
than 15 categories or scale points, it is referenced in PRELIS as ordinal (OR), whereas a variable with 15 or more categories is referenced as

Y102005.indb 19

3/22/10 3:25:17 PM

A Beginner’s Guide to Structural Equation Modeling

continuous (CO). This 15-point criterion allows Pearson correlation coefficient values to vary between +/–1.0. Variables with fewer distinct scale
points restrict the value of the Pearson correlation coefficient such that it
may only vary between +/–0.5. Other factors that affect the Pearson correlation coefficient are presented in this chapter and discussed further
in chapter 3.
2.2.3 Missing Data
The statistical analysis of data is affected by missing data values in variables. That is, not every subject has an actual value for every variable in
the dataset, as some values are missing. It is common practice in statistical packages to have default values for handling missing values. The
researcher has the options of deleting subjects who have missing values,
replacing the missing data values, or using robust statistical procedures
that accommodate for the presence of missing data.
The various SEM software handle missing data differently and have
different options for replacing missing data values. Table 2.1 lists many
of the various options for dealing with missing data. These options can
dramatically affect the number of subjects available for analysis, the
magnitude and direction of the correlation coefficient, or create problems
if means, standard deviations, and correlations are computed based on
different sample sizes. The Listwise deletion of cases and Pairwise deletion of cases are not always recommended options due to the possibility of losing a large number of subjects, thus dramatically reducing the
sample size. Mean substitution works best when only a small number
of missing values is present in the data, whereas regression imputation
provides a useful approach with a moderate amount of missing data.
In LISREL–PRELIS the expectation maximization (EM), Monte Carlo
Markov Chain (MCMC), and matching response pattern approaches
are recommended when larger amounts of data are missing at random.
TABLE 2.1
Options for Dealing with Missing Data
Listwise
Pairwise
Mean substitution
Regression imputation
Expectation
maximization (EM)
Matching response
pattern

Y102005.indb 20

Delete subjects with missing data on any variable
Delete subjects with missing data on each pair of variables used
Substitute the mean for missing values of a variable
Substitute a predicted value for the missing value of a variable
Find expected value based on expectation maximization
algorithm
Match cases with incomplete data to cases with complete data
to determine a missing value

3/22/10 3:25:17 PM

Data Entry and Data Editing Issues

More information about missing data is available in resources such as
Enders (2006), McKnight, McKnight, Sidani and Aurelio (2007), and
Peng, Harwell, Liou, and Ehman (2007). Davey and Savla (2010) have
more recently published an excellent book with SAS, SPSS, STATA, and
Mplus source programs to handle missing data in SEM in the context of
power analysis.
2.2.4 LISREL–PRELIS Missing Data Example
Imputation of missing values is possible for a single variable (Impute
Missing Values) or several variables simultaneously (Multiple Imputation)
by selecting Statistics from the tool bar menu. The Impute Missing Values
option uses the matching response pattern approach. The value to be substituted for the missing value of a single case is obtained from another
case (or cases) having a similar response pattern over a set of matching
variables. In data sets where missing values occur on more than one variable, you can use multiple imputation of missing values with mean substitution, delete cases, or leave the variables with defined missing values
as options in the dialog box. In addition, the Multiple Imputation option
uses either the expectation maximization algorithm (EM) or Monte Carlo
Markov Chain (MCMC, generating random draws from probability distributions via Markov chains) approaches to replacing missing values
across multiple variables.
We present an example from LISREL–PRELIS involving the cholesterol levels for 28 patients treated for heart attacks. We assume the data
to be missing at random (MAR) with an underlying multivariate normal
distribution. Cholesterol levels were measured after 2 days (VAR1), after
4 days (VAR2), and after 14 days (VAR3), but were only complete for 19
of the 28 patients. The data are shown from the PRELIS System File,
chollev.psf. The PRELIS system file was created by selecting File, Import
Data, and selecting the raw data file chollev.raw located in the Tutorial
folder [C:\LISREL 8.8 Student Examples\Tutorial]. We must know the number of variables in the raw data file. We must also select Data, then Define
Variables, and then select −9.00 as the missing value for the VAR3 variable [Optionally, right mouse click on VAR1 in the PRELIS chollev file].

Y102005.indb 21

3/22/10 3:25:18 PM

Y102005.indb 22

A Beginner’s Guide to Structural Equation Modeling

3/22/10 3:25:18 PM

Data Entry and Data Editing Issues

We now click on Statistics on the tool bar menu and select Impute
Missing Values from the pull-down menu.

We next select Output Options and save the transformed data in a new
PRELIS system file cholnew.psf, and output the new correlation matrix,
mean, and standard deviation files.

Y102005.indb 23

3/22/10 3:25:19 PM

A Beginner’s Guide to Structural Equation Modeling

We should examine our data both before (Table 2.2) and after (Table 2.3)
imputation of missing values. Here, we used the matching response pattern method. This comparison provides us with valuable information
about the nature of the missing data.
We can also view our new transformed PRELIS System File, cholnew.psf,
to verify that the missing values were in fact replaced; for example, VAR3
has values replaced for Case 2 = 204, Case 4 = 142, Case 5 = 182, Case 10 =
280, and so on.

Y102005.indb 24

3/22/10 3:25:19 PM

Data Entry and Data Editing Issues

TABLE 2.2
Data Before Imputation of Missing Values
Number of Missing Values per Variable
VAR1
-------------0

VAR2
------------0

VAR3
-----------9

Distribution of Missing Values
Total Sample Size = 28
Number of Missing Values
Number of Cases

0 1
19 9

Effective Sample Sizes
Univariate (in Diagonal) and Pairwise Bivariate (off
Diagonal)

VAR1
VAR2
VAR3

VAR1
-------------28
28
19

VAR2
-------------

VAR3
------------

28
19

Percentage of Missing Values
Univariate (in Diagonal) and Pairwise Bivariate (off
Diagonal)

VAR1
VAR2
VAR3

VAR1
-------------0.00
0.00
32.14

VAR2
-------------

VAR3
------------

0.00
32.14

32.14

VAR1
-------------1.000
0.673
0.395

VAR2
-------------

VAR3
------------

1.000
0.665

1.000

VAR1
-------------253.929

VAR2
------------230.643

VAR3
-----------221.474

VAR2
------------46.967

VAR3
-----------43.184

Correlation Matrix

VAR1
VAR2
VAR3
Means

Standard Deviations
VAR1
-------------47.710

Y102005.indb 25

3/22/10 3:25:20 PM

A Beginner’s Guide to Structural Equation Modeling

TABLE 2.3
Data After Imputation of Missing Values
Number of Missing Values per Variable
VAR1
-----------------0
Imputations for VAR3
Case 2 imputed with value
Case 4 imputed with value
Case 5 imputed with value
Case 10 imputed with value
Case 13 imputed with value
Case 16 imputed with value
Case 18 imputed with value
Case 23 imputed with value
Case 25 imputed with value

204
142
182
280
248
256
216
188
256

VAR2
---------------0

(Variance
(Variance
(Variance
(Variance
(Variance
(Variance
(Variance
(Variance
(Variance

Ratio
Ratio
Ratio
Ratio
Ratio
Ratio
Ratio
Ratio
Ratio

=
=
=
=
=
=
=
=
=

VAR3
---------------9

0.000),
0.000),
0.000),
0.000),
0.000),
0.000),
0.000),
0.000),
0.000),

NM=
NM=
NM=
NM=
NM=
NM=
NM=
NM=
NM=

1
1
1
1
1
1
1
1
1

Number of Missing Values per Variable After Imputation
VAR1
-----------------0

VAR2
---------------0

VAR3
---------------0

VAR1
-----------------1.000
0.673
0.404

VAR2
----------------

VAR3
----------------

1.000
0.787

1.000

VAR1
-----------------253.929

VAR2
---------------230.643

VAR3
---------------220.714

VAR2
---------------46.967

VAR3
---------------42.771

Total Sample Size = 28
Correlation Matrix

VAR1
VAR2
VAR3
Means

Standard Deviations
VAR1
-----------------47.710

We have noticed that selecting matching variables with a higher correlation to the variable with missing values provides better imputed
values for the missing data. We highly recommend comparing any analyses before and after the replacement of missing data values to fully
understand the impact missing data values have on the parameter estimates and standard errors. LISREL–PRELIS also permits replacement

Y102005.indb 26

3/22/10 3:25:20 PM

Data Entry and Data Editing Issues

of missing values using the EM and MCMC approaches, which may be
practical when matching sets of variables are not possible. A comparison
of EM and MCMC is also warranted in multiple imputations to determine the effect of using a different algorithm on the replacement of missing values.
2.2.5 Outliers
Outliers or influential data points can be defined as data values that are
extreme or atypical on either the independent (X variables) or dependent
(Y variables) variables or both. Outliers can occur as a result of observation errors, data entry errors, instrument errors based on layout or instructions, or actual extreme values from self-report data. Because outliers
affect the mean, the standard deviation, and correlation coefficient values,
they must be explained, deleted, or accommodated by using robust statistics. Sometimes, additional data will need to be collected to fill in the
gap along either the Y or X axes. LISREL–PRELIS has outlier detection
methods available that include the following: box plot display, scatterplot,
histogram, and frequency distributions.
2.2.6 Linearity
Some statistical techniques, such as SEM, assume that the variables are linearly related to one another. Thus, a standard practice is to visualize the
coordinate pairs of data points of two continuous variables by plotting the
data in a scatterplot. These bivariate plots depict whether the data are linearly increasing or decreasing. The presence of curvilinear data reduces the
magnitude of the Pearson correlation coefficient, even resulting in the presence of a zero correlation. Recall that the Pearson correlation value indicates
the magnitude and direction of the linear relationships between two variables. Figure 2.1 shows the importance of visually displaying the bivariate
data scatterplot.

Figure 2.1
Left: correlation is linear. Right: correlation is nonlinear.

Y102005.indb 27

3/22/10 3:25:20 PM

A Beginner’s Guide to Structural Equation Modeling

Table 2.4
Data Transformation Types
y = ln(x) or y = log10(x) or
y = ln(x+0.5)
y = sqrt(x)
y = arcsin((x + 0.375)/(n + 0.75))
y = 1/x
y = logit(x) = ln(x/(1 – x))
y = normit(x)
y = probit(x) = 5 + normit(x)

Useful with clustered data or cases where the standard
deviation increases with the mean
Useful with Poisson counts
Useful with binomial proportions [0.2 < p = x/n < 0.8]
Useful with gamma-distributed x variable
Useful with binomial proportions x = p
Quantile of normal distribution for standardized x
Most useful to resolve nonnormality of data

Note: probit(x) is same as normit(x) plus 5 to avoid negative values.

2.2.7 Nonnormality
In basic statistics, several transformations are given to handle issues with
nonnormal data. Some of these common transformations are in Table 2.4.
Inferential statistics often rely on the assumption that the data are normally distributed. Data that are skewed (lack of symmetry) or more frequently occurring along one part of the measurement scale will affect the
variance–covariance among variables. In addition, kurtosis (peakedness)
in data will impact statistics. Leptokurtic data values are more peaked than
the normal distribution, whereas platykurtic data values are flatter and
more dispersed along the X axis, but have a consistent low frequency on
the Y axis—that is, the frequency distribution of the data appears more
rectangular in shape.
Nonnormal data can occur because of the scaling of variables (e.g.,
ordinal rather than interval) or the limited sampling of subjects. Possible
solutions for skewness are to resample more participants or to perform a
linear transformation as outlined above. Our experience is that a probit
data transformation works best in correcting skewness. Kurtosis in data
is more difficult to resolve; some possible solutions in LISREL–PRELIS
include additional sampling of subjects, or the use of bootstrap methods, normalizing scores, or alternative methods of estimation (e.g., WLS
or ADF).
The presence of skewness and kurtosis can be detected in LISREL–
PRELIS using univariate tests, multivariate tests, and measures of skewness and kurtosis that are available in the pull-down menus or output.
One recommended method of handling nonnormal data is to use an
asymptotic covariance matrix as input along with the sample covariance
matrix in the LISREL–PRELIS program, as follows:

Y102005.indb 28

3/22/10 3:25:21 PM

Data Entry and Data Editing Issues

LISREL
CM = boy.cov
AC = boy.acm
SIMPLIS
Covariance matrix from file boy.cov
Asymptotic covariance matrix from file boy.acm

We can use the asymptotic covariance matrix in two different ways: (a) as a
weight matrix when specifying the method of estimation as weighted least
squares (WLS), and (b) as a weight matrix that adjusts the normal-theory
weight matrix to correct for bias in standard errors and fit statistics. The
appropriate moment matrix in PRELIS, using OUTPUT OPTIONS, must
be selected before requesting the calculation of the asymptotic covariance
matrix.
PRELIS recognizes data as being continuous (CO), ordinal (OR), or
classes (CL), that is gender (boy, girl). Different correlations are possible
depending upon the level of measurement. A variance–covariance matrix
with continuous variables would use Pearson correlations, while ordinal
variables would use Tetrachoric correlations. If skewed nonnormal data
is present, then consider a linear transformation using Probit. In SEM,
researchers typically output and use an asymptotic variance–covariance
matrix. When using a PRELIS data set, consider the normal score option
in the menu to correct for nonnormal variables.

2.3 Summary
Structural equation modeling is a correlation research method; therefore,
the measurement scale, restriction of range in the data values, missing
data, outliers, nonlinearity, and nonnormality of data affect the variance–
covariance among variables and thus can impact the SEM analysis.
Researchers should use the built-in menu options to examine, graph, and
test for any of these problems in the data prior to conducting any SEM
model analysis. Basically, researchers should know their data characteristics. Data screening is a very important first step in structural equation
modeling. The next chapter illustrates in more detail issues related to the
use of correlation and variance–covariance in SEM models. There, we
provide specific examples to illustrate the importance of topics covered
in this chapter. A troubleshooting box summarizing these issues is provided in Box 2.1.

Y102005.indb 29

3/22/10 3:25:21 PM

A Beginner’s Guide to Structural Equation Modeling

Box 2.1
Issue
Measurement
scale
Restriction of
range
Missing data

Outliers

Linearity

Nonnormality

Troubleshooting Tips

Suggestions
Need to take the measurement scale of the variables into account
when computing statistics such as means, standard deviations, and
correlations.
Need to consider range of values obtained for variables, as
restricted range of one or more variables can reduce the
magnitude of correlations.
Need to consider missing data on one or more subjects for one or
more variables as this can affect SEM results. Cases are lost with
listwise deletion, pairwise deletion is often problematic (e.g.,
different sample sizes), and thus modern imputation methods are
recommended.
Need to consider outliers as they can affect statistics such as
means, standard deviations, and correlations. They can either be
explained, deleted, or accommodated (using either robust
statistics or obtaining additional data to fill-in). Can be detected
by methods such as box plots, scatterplots, histograms or
frequency distributions.
Need to consider whether variables are linearly related, as
nonlinearity can reduce the magnitude of correlations. Can be
detected by scatterplots. Can be dealt with by transformations or
deleting outliers.
Need to consider whether the variables are normally distributed,
as nonnormality can affect resulting SEM statistics. Can be
detected by univariate tests, multivariate tests, and skewness and
kurtosis statistics. Can be dealt with by transformations,
additional sampling, bootstrapping, normalizing scores, or
alternative methods of estimation.

Exercises
1. LISREL uses which command to import data sets?
a. File, then Export Data
b. File, then Open
c. File, then Import Data
d. File, then New
2. Define the following levels of measurement.
a. Nominal
b. Ordinal
c. Interval
d. Ratio
3. Mark each of the following statements true (T) or false (F).
a. LISREL can deal with missing data.
b. PRELIS can deal with missing data.

Y102005.indb 30

3/22/10 3:25:21 PM

Data Entry and Data Editing Issues

c.
d.

LISREL can compute descriptive statistics.
PRELIS can compute descriptive statistics.

4. Explain how each of the following affects statistics:
a. Restriction of range
b. Missing data
c. Outliers
d. Nonlinearity
e. Nonnormality

References
Anderson, N. H. (1961). Scales and statistics: Parametric and non-parametric.
Psychological Bulletin, 58, 305–316.
Davey, A., & Savla, J. (2009). Statistical power analysis with missing data: A structural
equation modeling approach. Routledge, Taylor & Francis Group: New York.
Enders, C. K. (2006). Analyzing structural equation models with missing data. In
G.R. Hancock & R.O. Mueller (Eds.), Structural equation modeling: A second
course (pp. 313–342). Greenwich, CT: Information Age.
Jöreskog, K. G., & Sörbom, D. (1996). PRELIS2: User’s reference guide. Lincolnwood,
IL: Scientific Software International.
McKnight, P. E., McKnight, K. M., Sidani, S., & Aurelio, J. F. (2007). Missing data: A
gentle introduction. New York: Guilford.
Peng, C.-Y. J., Harwell, M., Liou, S.-M., & Ehman, L. H. (2007). Advances in missing
data methods and implications for educational research. In S.S. Sawilowsky
(Ed.), Real data analysis. Charlotte: Information Age.
Stevens, S. S. (1946). On the theory of scales of measurement. Science, 103,
677–680.

Y102005.indb 31

3/22/10 3:25:21 PM

3
Correlation

Key Concepts
Types of correlation coefficients
Factors affecting correlation
Correction for attenuation
Nonpositive definite matrices
Bivariate, part, and partial correlation
Suppressor variable
Covariance and causation
In chapter 2 we considered a number of data preparation issues in structural equation modeling. In this chapter, we move beyond data preparation in describing the important role that correlation (covariance) plays
in SEM. We also include a discussion of a number of factors that affect
correlation coefficients as well as the assumptions and limitations of correlation methods in structural equation modeling.

3.1 Types of Correlation Coefficients
Sir Francis Galton conceptualized the correlation and regression procedure for examining covariance in two or more traits, and Karl Pearson
(1896) developed the statistical formula for the correlation coefficient and
regression based on his suggestion (Crocker & Algina, 1986; Ferguson &
Takane, 1989; Tankard, 1984). Shortly thereafter, Charles Spearman (1904)
used the correlation procedure to develop a factor analysis technique.
The correlation, regression, and factor analysis techniques have for many
decades formed the basis for generating tests and defining constructs.
Today, researchers are expanding their understanding of the roles that
correlation, regression, and factor analysis play in theory and construct
33

Y102005.indb 33

3/22/10 3:25:21 PM

A Beginner’s Guide to Structural Equation Modeling

definition to include latent variable, covariance structure, and confirmatory factor measurement models.
The relationships and contributions of Galton, Pearson, and Spearman
to the field of statistics, especially correlation, regression, and factor analysis, are quite interesting (Tankard, 1984). In fact, the basis of association
between two variables—that is, correlation or covariance—has played a
major role in statistics. The Pearson correlation coefficient provides the
basis for point estimation (test of significance), explanation (variance
accounted for in a dependent variable by an independent variable), prediction (of a dependent variable from an independent variable through linear regression), reliability estimates (test–retest, equivalence), and validity
(factorial, predictive, concurrent).
The Pearson correlation coefficient also provides the basis for establishing and testing models among measured and/or latent variables. The
partial and part correlations further permit the identification of specific
bivariate relationships between variables that allow for the specification
of unique variance shared between two variables while controlling for the
influence of other variables. Partial and part correlations can be tested for
significance, similar to the Pearson correlation coefficient, by simply using
the degrees of freedom, n – 2, in the standard correlation table of significance values (Table A.3) or an F test in multiple regression which tests the
difference in R 2 values between full and restricted models (Table A.5).
Although the Pearson correlation coefficient has had a major impact in
the field of statistics, other correlation coefficients have emerged depending upon the level of variable measurement. Stevens (1968) provided the
properties of scales of measurement that have become known as nominal,
ordinal, interval, and ratio. The types of correlation coefficients developed
for these various levels of measurement are categorized in Table 3.1.
Table 3.1
Types of Correlation Coefficients
Correlation Coefficient
Pearson product‑moment
Spearman rank, Kendall’s tau
Phi, contingency
Point biserial
Gamma, rank biserial
Biserial
Polyserial
Tetrachoric
Polychoric
a

Y102005.indb 34

Level of Measurement
Both variables interval
Both variables ordinal
Both variables nominal
One variable interval, one variable dichotomous
One variable ordinal, one variable nominal
One variable interval, one variable artificiala
One variable interval, one variable ordinal with
underlying continuity
Both variables dichotomous (nominal artificiala)
Both variables ordinal with underlying continuities

Artificial refers to recoding variable values into a dichotomy.

3/22/10 3:25:21 PM

Correlation

Many popular computer programs, for example, SAS and SPSS, typically do not compute all of these correlation types. Therefore, you may
need to check a popular statistics book or look around for a computer program that will compute the type of correlation coefficient you need—for
example, the phi and point-biserial coefficient are not readily available. In
SEM analyses, the Pearson coefficient, tetrachoric or polychoric (for several
ordinal variable pairs) coefficient, and biserial or polyserial (for several
continuous and ordinal variable pairs) coefficient are typically used (see
PRELIS for the use of Kendall’s tau-c or tau-b, and canonical correlation).
LISREL permits mixture models, which use variables with both ordinal and
interval-ratio levels of measurement (chapter 15). Although SEM software
programs are now demonstrating how mixture models can be analyzed,
the use of variables with different levels of measurement has traditionally
been a problem in the field of statistics—for example, multiple regression
and multivariate statistics.

3.2 Factors Affecting Correlation Coefficients
Given the important role that correlation plays in structural equation
modeling, we need to understand the factors that affect establishing relationships among multivariable data points. The key factors are the level
of measurement, restriction of range in data values (variability, skewness,
kurtosis), missing data, nonlinearity, outliers, correction for attenuation,
and issues related to sampling variation, confidence intervals, effect size,
significance, sample size, and power.
3.2.1 Level of Measurement and Range of Values
Four types or levels of measurement typically define whether the characteristic or scale interpretation of a variable is nominal, ordinal, interval, or
ratio (Stevens, 1968). In structural equation modeling, each of these types
of scaled variables can be used. However, it is not recommended that they
be included together or mixed in a correlation (covariance) matrix. Instead,
the PRELIS data output option should be used to save an asymptotic covariance matrix for input along with the sample variance-covariance matrix
into a LISREL or SIMPLIS program.
Initially, SEM required variables measured at the interval or ratio level
of measurement, so the Pearson product-moment correlation coefficient
was used in regression, path, factor, and structural equation modeling.
The interval or ratio scaled variable values should also have a sufficient
range of score values to introduce variance (15 or more scale points). If the

Y102005.indb 35

3/22/10 3:25:22 PM

A Beginner’s Guide to Structural Equation Modeling

range of scores is restricted, the magnitude of the correlation value is
decreased. Basically, as a group of subjects become more homogeneous,
score variance decreases, reducing the correlation value between the variables. So, there must be enough variation in scores to allow a correlation
relationship to manifest itself between variables. Variables with fewer
than 15 categories are treated as ordinal variables in LISREL–PRELIS, so
if you are assuming continuous interval-level data, you will need to check
whether the variables meet this condition. Also, the use of the same scale
values for variables can help in the interpretation of results and/or relative comparison among variables. The meaningfulness of a correlation
relationship will depend on the variables employed; hence, your theoretical perspective is very important. You may recall from your basic statistics
course that a spurious correlation is possible when two sets of scores correlate significantly, but their relationship is not meaningful or substantive
in nature.
If the distributions of variables are widely divergent, correlation can
also be affected, and so several data transformations are suggested by
Ferguson and Takane (1989) to provide a closer approximation to a normal, homogeneous variance for skewed or kurtotic data. Some possible
transformations are the square root transformation (sqrt X), the logarithmic transformation (log X), the reciprocal transformation (1/X), and the
arcsine transformation (arcsin X). The probit transformation appears to be
most effective in handling univariate skewed data.
Consequently, the type of scale used and the range of values for the
measured variables can have profound effects on your statistical analysis
(in particular, on the mean, variance, and correlation). The scale and range
of a variable’s numerical values affects statistical methods, and this is no
different in structural equation modeling. The PRELIS program is available to provide tests of normality, skewness, and kurtosis on variables
and to compute an asymptotic covariance matrix for input into LISREL if
required. The use of normal scores is also an option in PRELIS.
3.2.2 Nonlinearity
The Pearson correlation coefficient indicates the degree of linear relationship between two variables. It is possible that two variables can indicate no
correlation if they have a curvilinear relationship. Thus, the extent to which
the variables deviate from the assumption of a linear relationship will affect
the size of the correlation coefficient. It is therefore important to check for
linearity of the scores; the common method is to graph the coordinate data
points in a scatterplot. The linearity assumption should not be confused
with recent advances in testing interaction in structural equation models
discussed in chapter 16. You should also be familiar with the eta coefficient
as an index of nonlinear relationship between two variables and with the

Y102005.indb 36

3/22/10 3:25:22 PM

Correlation

testing of linear, quadratic, or cubic effects. Consult an intermediate statistics text, for example, Lomax (2007) to review these basic concepts.
The heuristic data sets in Table 3.2 will demonstrate the dramatic effect
a lack of linearity has on the Pearson correlation coefficient value. In the
first data set, the Y values increase from 1 to 10, and the X values increase
from 1 to 5, then decrease from 5 to 1 (nonlinear). The result is a Pearson
correlation coefficient of r = 0; although a nonlinear relationship does exist
in the data, it is not indicated by the Pearson correlation coefficient. The
restriction of range in values can be demonstrated using the fourth heuristic data set in Table 3.2. The Y values only range between 3 and 7, and
the X values only range from 1 to 4. The Pearson correlation coefficient is
also r = 0 for these data. The fifth data set indicates how limited sampling
can affect the Pearson coefficient. In these sample data, only three pairs
of scores are sampled, and the Pearson correlation is r = –1.0, or perfectly
negatively correlated.
TABLE 3.2
Heuristic Data Sets
Nonlinear Data

Missing Data

1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00

1.00
2.00
3.00
4.00
5.00
5.00
4.00
3.00
2.00
1.00

8.00
7.00
8.00
5.00
4.00
5.00
3.00
5.00
3.00
2.00

6.00
5.00
4.00
2.00
3.00
2.00
3.00
4.00
1.00
2.00

8.00
7.00
8.00
5.00
4.00
5.00
3.00
5.00
3.00
2.00

—
5.00
—
2.00
3.00
2.00
3.00
—
1.00
2.00

Range of Data
Y
3.00
3.00
4.00
4.00
5.00
5.00
6.00
6.00
7.00
7.00

Y102005.indb 37

Complete Data

X
1.00
2.00
3.00
4.00
1.00
2.00
3.00
4.00
1.00
2.00

Sampling Effect
Y
8.00
9.00
10.00

X
3.00
2.00
1.00

3/22/10 3:25:22 PM

A Beginner’s Guide to Structural Equation Modeling

3.2.3 Missing Data
A complete data set is also given in Table 3.2 where the Pearson correlation coefficient is r = .782, p = .007, for n = 10 pairs of scores. If missing
data were present, the Pearson correlation coefficient would drop to r =
.659, p = .108, for n = 7 pairs of scores. The Pearson correlation coefficient
changes from statistically significant to not statistically significant. More
importantly, in a correlation matrix with several variables, the various
correlation coefficients could be computed on different sample sizes. If
we used listwise deletion of cases, then any variable in the data set with
a missing value would cause a subject to be deleted, possibly causing a
substantial reduction in our sample size, whereas pairwise deletion of cases
would result in different sample sizes for our correlation coefficients in
the correlation matrix.
Researchers have examined various aspects of how to handle or treat
missing data beyond our introductory example using a small heuristic
data set. One basic approach is to eliminate any observations where some
of the data are missing, listwise deletion. Listwise deletion is not recommended because of the loss of information on other variables, and statistical estimates are based on reduced sample size. Pairwise deletion excludes
data only when they are missing on the pairs of variables selected for
analysis. However, this could lead to different sample sizes for the different correlations and related statistical estimates. A third approach, data
imputation, replaces missing values with an estimate, for example, the
mean value on a variable for all subjects who did not report any data for
that variable (Beale & Little, 1975; also see chapter 2).
Missing data can arise in different ways (Little & Rubin, 1987, 1990).
Missing completely at random (MCAR) implies that data on variable X are
missing unrelated statistically to the values that have been observed
for other variables as well as X. Missing at random (MAR) implies that
data values on variable X are missing conditional on other variables,
but are unrelated to the values of X. A third situation, nonignorable data,
implies probabilistic information about the values that would have been
observed. For MCAR data, mean substitution yields biased variance and
covariance estimates, whereas listwise and pairwise deletion methods
yield consistent solutions. For MAR data, mean substitution, listwise,
and pairwise deletion methods produce biased results. When missing
data are nonignorable, all approaches yield biased results. It would be
prudent for the researcher to investigate how parameter estimates are
affected by the use or nonuse of a data imputation method. A few references are provided to give a more detailed understanding of missing data (Arbuckle, 1996; Enders, 2006; McKnight, McKnight, Sidani &
Aurelio, 2007; Peng, Harwell, Liou & Ehman, 2007; Wothke, 2000; Davey
& Savla, 2009).

Y102005.indb 38

3/22/10 3:25:22 PM

Correlation

3.2.4 Outliers
The Pearson correlation coefficient can be drastically affected by a single outlier on X or Y. For example, the two data sets in Table 3.3 indicate
a Y = 27 value (Set A) versus a Y = 2 value (Set B) for the last subject. In
the first set of data, r = .524, p = .37, whereas in the second set of data,
r = –.994, p = .001. Is the Y = 27 data value an outlier based on limited
sampling or is it a data entry error? A large body of research has been
undertaken to examine how different outliers on X, Y, or both X, and
Y affect correlation relationships, and how to better analyze the data
using robust statistics (Anderson & Schumacker, 2003; Ho & Naugher,
2000; Huber, 1981; Rousseeuw & Leroy, 1987; Staudte & Sheather, 1990).
TABLE 3.3
Outlier Data Sets
Set A

Set B

1
2
3
4
5

9
7
5
3
27

1
2
3
4
5

9
7
5
3
2

3.2.5 Correction for Attenuation
A basic assumption in psychometric theory is that observed data contain measurement error. A test score (observed data) is a function of a true score and
measurement error. A Pearson correlation coefficient will have different values, depending on whether it was computed with observed scores or the true
scores where measurement error has been removed. The Pearson correlation
coefficient can be corrected for attenuation or unreliable measurement error in
scores, thus yielding a true score correlation; however, the corrected correlation coefficient can become greater than 1.0! Low reliability in the independent and/or dependent variables, coupled with a high correlation between
the independent and dependent variable, can result in correlations greater
than 1.0. For example, given a correlation of r = .90 between the observed
scores on X and Y, the Cronbach alpha reliability coefficient of .60 for X scores,
and the Cronbach alpha reliability coefficient of .70 for Y scores, the Pearson
correlation coefficient, corrected for attenuation (r*) , is greater than 1.0:
r * xy =

Y102005.indb 39

rxy
rxx ryy

.90
.60(.70)

.90
= 1.389
.648

3/22/10 3:25:23 PM

A Beginner’s Guide to Structural Equation Modeling

When this happens, a nonpositive definite error message occurs stopping
the SEM program.
3.2.6 Nonpositive Definite Matrices
Correlation coefficients greater than 1.0 in a correlation matrix cause the
correlation matrix to be nonpositive definite. In other words, the solution is
not admissible, indicating that parameter estimates cannot be computed.
Correction for attenuation is not the only situation that causes nonpositive matrices to occur (Wothke, 1993). Sometimes the ratio of covariance
to the product of variable variances yields correlations greater than 1.0.
The following variance–covariance matrix is nonpositive definite because
it contains a correlation coefficient greater than 1.0 between the Relations
and Attribute latent variables (denoted by an asterisk):
Variance–Covariance Matrix
Task
Relations
Management
Attribute

1.043
.994
.892
1.065

1.079
.905
1.111

.924
.969

1.12

1.000
.937
.908
.985

1.000
.906
1.010*

1.000
.951

1.000

Correlation Matrix
Task
Relations
Management
Attribute

Nonpositive definite covariance matrices occur when the determinant of
the matrix is zero or the inverse of the matrix is not possible. This can
be caused by correlations greater than 1.0, linear dependency among the
observed variables, multicollinearity among the observed variables, a
variable that is a linear combination of other variables, a sample size less
than the number of variables, the presence of a negative or zero variance
(Heywood Case), variance–covariance (correlation) values outside the
permissible range, for example, correlation beyond +/−1.0, and bad start
values in the user-specified model. A Heywood case also occurs when the
communality estimate is greater than 1.0. Possible solutions to resolve
this error are to reduce communality or fix communality to less than 1.0,
extract a different number of factors (possibly by dropping paths), rescale
observed variables to create a more linear relationship, or eliminate a bad
observed variable that indicates linear dependency or multicollinearity.
Regression, path, factor, and structural equation models mathematically
solve a set of simultaneous equations typically using ordinary least squares

Y102005.indb 40

3/22/10 3:25:23 PM

Correlation

(OLS) estimates as initial estimates of coefficients in the model. However,
these initial estimates or coefficients are sometimes distorted or too different from the final admissible solution. When this happens, more reasonable start values need to be chosen. It is easy to see from the basic regression
coefficient formula that the correlation coefficient value and the standard
deviation values of the two variables affect the initial OLS estimates:
 sy 
b = rxy   .
 sx 
3.2.7 Sample Size
A common formula used to determine sample size when estimating means
of variables was given by McCall (1982): n = (Z s/e)2, where n is the sample
size needed for the desired level of precision, e is the effect size, Z is the
confidence level, and s is the population standard deviation of scores
(s can be estimated from prior research studies, test norms, or the range of
scores divided by 6). For example, given a random sample of ACT scores
from a defined population with a standard deviation of 100, a desired confidence level of 1.96 (which corresponds to a .05 level of significance), and
an effect size of 20 (difference between sampled ACT mean and population ACT mean), the sample size needed would be [1.96 (100)/20)]2 = 96.
In structural equation modeling, however, the researcher often requires
a much larger sample size to maintain power and obtain stable parameter
estimates and standard errors. The need for larger sample sizes is also
due in part to the program requirements and the multiple observed variables used to define latent variables. Hoelter (1983) proposed the critical
N statistic, which indicates the sample size needed to obtain a chi-square
value that would reject the null hypothesis in a structural equation model.
The required sample size and power estimates that provide a reasonable
indication of whether a researcher’s data fits their theoretical model or to
estimate parameters is discussed in more detail in chapter 5.
SEM software programs estimate coefficients based on the user-specified
theoretical model, or implied model, but also must work with the saturated and independence models. A saturated model is the model with all
parameters indicated, while the independence model is the null model or
model with no parameters estimated. A saturated model with p observed
variables has p (p + 3)/2 free parameters [Note: Number of independent
elements in the symmetric covariance matrix = p(p + 1)/2. Number of
means = p, so total number of independent elements = p (p + 1)/2 + p = p
(p + 3)/2]. For example, with 10 observed variables, 10(10 + 3)/2 = 65 free
parameters. If the sample size is small, then there is not enough information to estimate parameters in the saturated model for a large number of
variables. Consequently, the chi-square fit statistic and derived statistics

Y102005.indb 41

3/22/10 3:25:23 PM

A Beginner’s Guide to Structural Equation Modeling

such as Akaike’s Information Criterion (AIC) and the root-mean-square
error of approximation (RMSEA) cannot be computed. In addition, the fit
of the independence model is required to calculate other fit indices such
as the Comparative Fit Index (CFI) and the Normed Fit Index (NFI).
Ding, Velicer, and Harlow (1995) located numerous studies (e.g.,
Anderson & Gerbing, 1988) that were in agreement that 100 to 150 subjects
is the minimum satisfactory sample size when conducting structural equation models. Boomsma (1982, 1983) recommended 400, and Hu, Bentler,
and Kano (1992) indicated that in some cases 5,000 is insufficient! Many
of us may recall rules of thumb in our statistics texts, for example, 10 subjects per variable or 20 subjects per variable. Costello and Osborne (2005)
demonstrated in their Monte Carlo study that 20 subjects per variable is
recommended for best practices in factor analysis. In our examination of
published SEM research, we have found that many articles used from 250
to 500 subjects, although the greater the sample size, the more likely it
is one can validate the model using cross-validation (see chapter 12). For
example, Bentler and Chou (1987) suggested that a ratio as low as five subjects per variable would be sufficient for normal and elliptical distributions
when the latent variables have multiple indicators and that a ratio of at
least 10 subjects per variable would be sufficient for other distributions.
Determination of sample size is now better understood in SEM modeling and further discussed in chapter 5.

3.3 Bivariate, Part, and Partial Correlations
The types of correlations indicated in Table 3.1 are considered bivariate correlations, or associations between two variables. Cohen & Cohen (1983), in
describing correlation research, further presented the correlation between
two variables controlling for the influence of a third variable. These correlations are referred to as part and partial correlations, depending upon how
variables are controlled or partialled out. Some of the various ways in which
three variables can be depicted are illustrated in Figure 3.1. The diagrams
illustrate different situations among variables where (a) all the variables are
uncorrelated (Case 1), (b) only one pair of variables is correlated (Cases 2
and 3), (c) two pairs of variables are correlated (Cases 4 and 5), and (d) all of
the variables are correlated (Case 6). It is obvious that with more than three
variables the possibilities become overwhelming. It is therefore important to
have a theoretical perspective to suggest why certain variables are correlated
and/or controlled in a study. A theoretical perspective is essential in specifying a model and forms the basis for testing a structural equation model.
The partial correlation coefficient measures the association between two
variables while controlling for a third variable, for example, the association

Y102005.indb 42

3/22/10 3:25:24 PM

Correlation

Z
CASE 1

CASE 2

Y
X

CASE 4

CASE 3

Y
Z

CASE 6

CASE 5

Figure 3.1
Possible three-variable relationships.

between age and reading comprehension, controlling for reading level.
Controlling for reading level in the correlation between age and comprehension partials out the correlation of reading level with age and the correlation of reading level with comprehension. Part correlation, in contrast,
is the correlation between age and comprehension with reading level controlled for, where only the correlation between comprehension and reading level is removed before age is correlated with comprehension.
Whether a part or partial correlation is used depends on the specific
model or research question. Convenient notation helps distinguish these
two types of correlations (1 = age, 2 = comprehension, 3 = reading level):
partial correlation, r12.3, part correlation, r 1(2.3) or r 2(1.3). Different correlation values are computed depending on which variables are controlled
or partialled out. For example, using the correlations in Table 3.4, we
can compute the partial correlation coefficient r 12.3 (correlation between
age and comprehension, controlling for reading level) as follows:
r 12.3 =
=

Y102005.indb 43

r12 − r13r23
2 )
(1 − r132 )(1 − r23
.45 − (.25) (.80)
= .43
[1 − (.25)2 ] [1 − (.80)2 ]

3/22/10 3:25:24 PM

A Beginner’s Guide to Structural Equation Modeling

TABLE 3.4
Correlation Matrix (n = 100)
Variable

Age

Comprehension

Reading Level

1. Age
2. Comprehension
3. Reading level

1.00
.45
.25

1.00
.80

1.00

Notice that the partial correlation coefficient should be smaller in magnitude than the Pearson product‑moment correlation between age and comprehension, which is r12 = .45. If the partial correlation coefficient is not
smaller than the Pearson product-moment correlation, then a suppressor
variable may be present (Pedhazur, 1997). A suppressor variable correlates
near zero with a dependent variable but correlates significantly with other
predictor variables. This correlation situation serves to control for variance
shared with predictor variables and not the dependent variable. The partial
correlation coefficient increases in magnitude once this effect is removed
from the correlation between two predictor variables with a criterion.
Partial correlations will be greater in magnitude than part correlations,
except when independent variables are zero correlated with the dependent variable; then, part correlations are equal to partial correlations.
The part correlation coefficient r1(2.3), or correlation between age and
comprehension where reading level is controlled for in comprehension
only, is computed as
r1(2.3) =

r12 − r13r23
1− r

2
23

.45 − (.25)(.80)
1 − .802

= .42 ,

or, in the case of correlating comprehension with age where reading level
is controlled for age only is
r2(1.3) =

r12 − r13r23
1 − r132

.45 − (.25)(.80)
1 − .252

= .26.

The correlation, whether zero‑order (bivariate), part, or partial can be
tested for significance, interpreted as variance accounted for by squaring
each coefficient, and diagrammed using Venn or Ballentine figures to conceptualize their relationships. In our example, the zero‑order relationships
among the three variables can be diagrammed as in Figure 3.2. However,
the partial correlation of age with comprehension controlling for reading
level would be r12.3 = .43, or area a divided by the combined area of a and
e [a/(a + e)]; see Figure 3.3. A part correlation of age with comprehension

Y102005.indb 44

3/22/10 3:25:25 PM

Correlation

Age and Comprehension

Age and Reading

Reading and Comprehension
Figure 3.2
Bivariate correlations.

while controlling for the correlation between reading level and comprehension would be r1(2.3) = .42, or just area a; see Figure 3.4.
These examples consider only controlling for one variable when correlating two other variables (partial), or controlling for the impact of one variable
on another before correlating with a third variable (part). Other higherorder part correlations and partial correlations are possible (e.g., r12.34, r12(3.4)),
but are beyond the scope of this book. Readers should refer to references for

Age

Comprehension

Reading
Figure 3.3
Partial correlation area.

Y102005.indb 45

3/22/10 3:25:26 PM

A Beginner’s Guide to Structural Equation Modeling

Age

Comprehension

Reading
Figure 3.4
Part correlation area.

a more detailed discussion of part and partial correlation (Cohen & Cohen,
1983; Pedhazur, 1997; Hinkle, Wiersma & Jurs, 2003; Lomax, 2007).

3.4 Correlation versus Covariance
The type of data matrix used for computations in structural equation modeling programs is a variance–covariance matrix. A variance–covariance matrix
is made up of variance terms on the diagonal and covariance terms on the
off-diagonal. If a correlation matrix is used as the input data matrix, most
of the computer programs by default convert it to a variance–covariance
matrix using the standard deviations of the variables, unless specified
otherwise. The researcher has the option to input raw data, a correlation
matrix, or a variance–covariance matrix. The correlation matrix provides
the option of using standardized or unstandardized variables for analysis
purposes. If a correlation matrix is input with a row of variable means
(although optional) and a row of standard deviations, then a variance–
covariance matrix is used with unstandardized output. If only a correlation matrix is input, the means and standard deviations, by default, are
set at 0 and 1, respectively, and standardized output is printed. When raw
data are input, a variance–covariance matrix is computed.
The number of distinct elements in a variance–covariance matrix S is
p(p + 1)/2, where p is the number of observed variables. For example, the
variance–covariance matrix for the following three variables, X, Y, and Z,
is as follows:

X
Y
Z

X
15.80
10.16
12.43

11.02
9.23

15.37

It has 3 (3 + 1)/2 = 6 distinct values: 3 variance and 3 covariance terms.

Y102005.indb 46

3/22/10 3:25:27 PM

Correlation

Correlation is computed using the variances and covariance among the
bivariate variables, using the following formula:
r=

2
sXY

sX2 * sY2

Dividing the covariance between two variables (covariance terms are the
off-diagonal values in the matrix) by the square root of the product of the
two variable variances (variances of variables are on the diagonal of the
matrix) yields the following correlations among the three variables:
rxy = 10.16/(15.80 * 11.02)1/2 = .77
rxz = 12.43/(15.80 * 15.37)1/2 = .80
ryz = 9.23/(11.02 * 15.37)1/2 = .71.
Structural equation software uses the variance–covariance matrix rather
than the correlation matrix because Boomsma (1983) found that the analysis of correlation matrices led to imprecise parameter estimates and standard errors of the parameter estimates in a structural equation model.
In SEM, incorrect estimation of the standard errors for the parameter
estimates could lead to statistically significant parameter estimates and
an incorrect interpretation of the model—that is, the parameter divided
by the standard error indicates a ratio statistic or T-value, which can be
compared to tabled critical t-values for statistical significance at different
alpha levels (Table A.2). Browne (1982), Jennrich and Thayer (1973), and
Lawley and Maxwell (1971) have suggested corrections for the standard
errors when correlations or standardized coefficients are used in SEM. In
general, a variance–covariance matrix should be used in structural equation modeling, although some SEM models require variable means, for
example, structured means models (see chapter 13).

3.5 Variable Metrics (Standardized versus Unstandardized)
Researchers have debated the use of unstandardized or standardized
variables (Lomax, 2007). The standardized coefficients are thought to be
sample specific and not stable across different samples because of changes
in the variance of the variables. The unstandardized coefficients permit
an examination of change across different samples. The standardized
coefficients are useful, however, in determining the relative importance
of each variable to other variables for a given sample. Other reasons for

Y102005.indb 47

3/22/10 3:25:27 PM

A Beginner’s Guide to Structural Equation Modeling

using standardized variables are that variables are on the same scale of
measurement, are more easily interpreted, and can easily be converted
back to the raw scale metric. In a SIMPLIS program, adding the command
LISREL OUTPUT SS SC provides a standardized solution (observed variables) and a completely standardized solution (observed variables and
latent variables).

3.6 Causation Assumptions and Limitations
As previously discussed, the Pearson correlation coefficient is limited by
the range of score values and the assumption of linearity, among other
things. Even if the assumptions and limitations of using the Pearson correlation coefficient are met, a cause-and-effect relationship still has not
been established. The following conditions are necessary for cause and
effect to be inferred between variables X and Y (Tracz, 1992): (a) temporal order (X precedes Y in time), (b) existence of covariance or correlation
between X and Y, and (c) control for other causes, for example, partial Z
out of X and Y.
These three conditions may not be present in the research design setting, and in such a case, only association rather than causation can be
inferred. However, if manipulative variables are used in the study, then
a researcher could change or manipulate one variable in the study and
examine subsequent effects on other variables, thereby determining
cause-and-effect relationships (Resta & Baker, 1972). In structural equation modeling, the amount of influence rather than a cause-and-effect
relationship is assumed and interpreted by direct, indirect, and total
effects among variables, which are explained in chapter 7 where we discuss path models.
Philosophical differences exist between assuming causal versus inference relationships among variables, and the resolution of these issues
requires a sound theoretical perspective. Bullock, Harlow, and Mulaik
(1994) provided an in-depth discussion of causation issues related to
structural equation modeling research. We feel that structural equation
models will evolve beyond model fit into the domain of model testing
as witnessed by the many new SEM model applications today. Model
testing rather than model fit can involve testing significance of parameters, parameter change, or other factors that affect the model outcome
values, and whose effects can be assessed. This approach, we believe, best
depicts a causal assumption. In addition, structural models in longitudinal research can depict changes in latent variables over time (Collins &
Horn, 1992). Pearl (2009) more recently has renewed a discussion about

Y102005.indb 48

3/22/10 3:25:27 PM

Correlation

causality and firmly believes it is not mystical or metaphysical, but rather
can be understood in terms of processes (models) that can be expressed in
mathematical expressions ready for computer analysis.

3.7 Summary
In this chapter, we have described some of the basic correlation concepts
underlying structural equation modeling. This discussion included various types of bivariate correlation coefficients, part and partial correlation,
variable metrics, factors affecting correlation, the assumptions required in
SEM, and causation versus inference debate in SEM modeling.
Most computer programs do not compute all the types of correlation
coefficients used in statistics, so the reader should refer to a standard statistics textbook for computational formulas and understanding (Hinkle,
Weirsma, & Jurs, 2003; Lomax, 2007). Structural equation modeling programs use a variance–covariance matrix, and include features to output
the type of matrices they use. In SEM, categorical and/or ordinal variables with underlying continuous latent-variable attributes have been
used with tetrachoric or polychoric correlations (Muthén, 1982, 1983, 1984;
Muthén & Kaplan, 1985). PRELIS has been developed to permit a correlation matrix of various types of correlations to be conditioned or converted
into an asymptotic covariance matrix for input into structural equation
modeling programs (Jöreskog & Sörbom, 1993). The use of various correlation coefficients and subsequent conversion into a variance–covariance
matrix will continue to play a major role in structural equation modeling,
especially given mixture models (see chapter 15).
The chapter also presented numerous factors that affect the Pearson
correlation coefficient, for example, restriction of range in the scores,
outliers, skewness, and nonnormality. SEM software also converts correlation matrices with standard deviations into a variance–covariance
matrix, but if attenuated correlations are greater than 1.0, a nonpositive
definite error message will occur because of an inadmissible solution.
Nonpositive definite error messages are all too common among beginners because they do not screen the data, thinking instead that structural equation modeling will be unaffected. Another major concern is
when OLS initial estimates lead to bad start values for the coefficients
in a model; however, changing the number of default iterations sometimes solves this problem. A troubleshooting box summarizes these
issues (see Box 3.1). In chapter 4, we begin to deal with the basic steps a
researcher takes in conducting SEM, which follows throughout the chapters in the book.

Y102005.indb 49

3/22/10 3:25:27 PM

A Beginner’s Guide to Structural Equation Modeling

Box-3.1
Issue

Troubleshooting Tips
Suggestions

Measurement
scale
Restriction of
range

Missing data

Outliers

Linearity

Correction for
attenuation
Nonpositive
definite
matrices

Sample size

Need to take the measurement scale of the variables into
account when computing correlations.
Need to consider range of values obtained for variables, as
restricted range of one or more variables can reduce the
magnitude of correlations. Can consider data transformations
for nonnormal data.
Need to consider missing data on one or more subjects for one or
more variables as this can affect SEM results. Cases are lost with
listwise deletion, pairwise deletion is often problematic (e.g.,
different sample sizes), and thus modern methods are
recommended.
Need to consider outliers as they can affect correlations. They
can either be explained, deleted, or accommodated (using
either robust statistics or obtaining additional data to fill-in).
Can be detected by methods such as box plots, scatterplots,
histograms or frequency distributions.
Need to consider whether variables are linearly related, as
nonlinearity can reduce the magnitude of correlations. Can be
detected by scatterplots and dealt with by transformations or
deleting outliers.
Less than perfect reliability on observed measures can reduce
the magnitude of correlations and lead to nonpositive definite
error message. Best to use multiple, high quality measures.
Can occur in a correlation or covariance matrix due to a
variable that a linear combination of other variables,
collinearity, sample size less than the number of variables,
negative or zero variances, correlations outside of the
permissible range, or bad start values. Solutions include
eliminating the bad variables, rescaling variables, and using
more reasonable starting values.
Small samples can reduce power and precision of parameter
estimates. At least 100 to 150 cases is necessary for smaller
models with well-behaved data.

Exercises
1. Given the Pearson correlation coefficients r12 = .6, r13 = .7, and
r23 = 4, compute the part and partial correlations r12.3 and r1(2.3).
2. Compare the variance explained in the bivariate, partial, and
part correlations of Exercise 1.
3. Explain causation and describe when a cause-and-effect relationship might exist.

Y102005.indb 50

3/22/10 3:25:27 PM

Correlation

4. Given the following variance-covariance matrix, compute the
Pearson correlation coefficients: rXY, rXZ, and rYZ:

X
Y
Z

15.80
10.16
12.43

11.02
9.23

15.37

References
Anderson, J. C., & Gerbing, D. W. (1988). Structural equation modeling in practice:
A review and recommended two step approach. Psychological Bulletin, 103,
411–423.
Anderson, C., & Schumacker, R. E. (2003). A comparison of five robust regression
methods with ordinary least squares regression: Relative efficiency, bias, and
test of the null hypothesis. Understanding Statistics, 2, 77–101.
Arbuckle, J. L. (1996). Full information estimation in the presence of incomplete
data. In G. A. Marcoulides and R. E. Schumacker (Eds.). Advanced structural
equation modeling (pp. 243–277). Mahwah, NJ: Lawrence Erlbaum Associates.
Beale, E. M. L., & Little, R. J. (1975). Missing values in multivariate analysis. Journal
of the Royal Statistical Society Series B, 37, 129–145.
Bentler, P. M., & Chou, C. (1987). Practical issues in structural equation modeling.
Sociological Methods and Research, 16, 78–117.
Boomsma, A. (1982). The robustness of LISREL against small sample sizes in
factor analysis models. In K. G. Jöreskog & H. Wold (Eds.), Systems under
indirect observation: Causality, structure, prediction (Part I) (pp. 149–173).
Amsterdam: North-Holland.
Boomsma, A. (1983). On the robustness of LISREL against small sample size and nonnormality. Amsterdam: Sociometric Research Foundation.
Browne, M. W. (1982). Covariance structures. In D. M. Hawkins (Ed.), Topics in applied
multivariate analysis (pp. 72–141). Cambridge: Cambridge University Press.
Bullock, H. E., Harlow, L. L., & Mulaik, S. A. (1994). Causation issues in structural
equation modeling. Structural Equation Modeling: A Multidisciplinary Journal,
1, 253–267.
Cohen, J., & Cohen, P. (1983). Applied multiple regression/correlation analysis for the
behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum Associates.
Collins, L. M., & Horn, J. L. (Eds.). (1992). Best methods for the analysis of change: Recent
advances, unanswered questions, future directions. Washington, DC: American
Psychological Association.
Costello, A. B., & Osborne, J. (2005). Best practices in exploratory factor analysis: four recommendations for getting the most from your analysis. Practical
Assessment Research & Evaluation, 10(7), 1–9.

Y102005.indb 51

3/22/10 3:25:27 PM

A Beginner’s Guide to Structural Equation Modeling

Crocker, L., & Algina, J. (1986). Introduction to classical and modern test theory. New
York: Holt, Rinehart & Winston.
Davey, A., & Savla, J. (2009). Statistical power analysis with missing data: A structural
equation modeling approach. Routledge, Taylor & Francis Group: New York.
Ding, L., Velicer, W. F., & Harlow, L. L. (1995). Effects of estimation methods, number of indicators per factor, and improper solutions on structural equation
modeling fit indices. Structural Equation Modeling: A Multidisciplinary Journal,
2, 119–143.
Enders, C. K. (2006). Analyzing structural equation models with missing data. In
G. R. Hancock & R. O. Mueller (Eds.), Structural equation modeling: A second
course (pp. 313–342). Greenwich, CT: Information Age.
Ferguson, G. A., & Takane, Y. (1989). Statistical analysis in psychology and education
(6th ed.). New York: McGraw-Hill.
Hinkle, D. E., Wiersma, W., & Jurs, S. G. (2003). Applied statistics for the behavioral
sciences (5th ed.). Boston: Houghton Mifflin.
Ho, K., & Naugher, J. R. (2000). Outliers lie: An illustrative example of identifying
outliers and applying robust methods. Multiple Linear Regression Viewpoints,
26(2), 2–6.
Hoelter, J. W. (1983). The analysis of covariance structures: Goodness-of-fit indices.
Sociological Methods and Research, 11, 325–344.
Hu, L., Bentler, P. M., & Kano, Y. (1992). Can test statistics in covariance structure
analysis be trusted? Psychological Bulletin, 112, 351–362.
Huber, P. J. (1981). Robust statistics. New York: Wiley.
Jennrich, R. I., & Thayer, D. T. (1973). A note on Lawley’s formula for standard
errors in maximum likelihood factor analysis. Psychometrika, 38, 571–580.
Jöreskog, K. G., & Sörbom, D. (1993). PRELIS2 user’s reference guide. Chicago:
Scientific Software International.
Lawley, D. N., & Maxwell, A. E. (1971). Factor analysis as a statistical method.
London: Butterworth.
Little, R. J., & Rubin, D. B. (1987). Statistical analysis with missing data. New
York: Wiley.
Little, R. J., & Rubin, D. B. (1990). The analysis of social science data with missing
values. Sociological Methods and Research, 18, 292–326.
Lomax, R. G. (2007). An introduction to statistical concepts (2nd ed.). Mahwah, NJ:
Lawrence Erlbaum Associates, Inc.
McCall, C. H., Jr. (1982). Sampling statistics handbook for research. Ames: Iowa State
University Press.
McKnight, P. E., McKnight, K. M., Sidani, S., & Aurelio, J. F. (2007). Missing data: A
gentle introduction. New York: Guilford.
Muthén, B. (1982). A structural probit model with latent variables. Journal of the
American Statistical Association, 74, 807–811.
Muthén, B. (1983). Latent variable structural equation modeling with categorical
data. Journal of Econometrics, 22, 43–65.
Muthén, B. (1984). A general structural equation model with dichotomous, ordered
categorical, and continuous latent variable indicators. Psychometrika, 49,
115–132.

Y102005.indb 52

3/22/10 3:25:28 PM

Correlation

Muthén, B., & Kaplan, D. (1985). A comparison of some methodologies for the factor analysis of non-normal Likert variables. British Journal of Mathematical and
Statistical Psychology, 38, 171–189.
Pearl, J. (2009). Causality: Models, reasoning, and inference (2nd edition). Cambridge
University Press: London.
Pearson, K. (1896). Mathematical contributions to the theory of evolution. Part 3.
Regression, heredity and panmixia. Philosophical Transactions, A, 187, 253–318.
Pedhazur, E. J. (1997). Multiple regression in behavioral research: Explanation and prediction (3rd ed.). Fort Worth: Harcourt Brace.
Peng, C.-Y. J., Harwell, M., Liou, S.-M., & Ehman, L. H. (2007). Advances in missing
data methods and implications for educational research. In S. S. Sawilowsky
(Ed.), Real Data Analysis. Charlotte: Information Age.
Resta, P. E., & Baker, R. L. (1972). Selecting variables for educational research.
Inglewood, CA: Southwest Regional Laboratory for Educational Research
and Development.
Rousseeuw, P. J., & Leroy, A. M. (1987). Robust regression and outlier detection. New
York: Wiley.
Spearman, C. (1904). The proof and measurement of association between two
things. American Journal of Psychology, 15, 72–101.
Staudte, R. G., & Sheather, S. J. (1990). Robust estimation and testing. New York: Wiley.
Stevens, S. S. (1968). Measurement, statistics, and the schempiric view. Science, 161,
849–856.
Tankard, J. W., Jr. (1984). The statistical pioneers. Cambridge, MA: Schenkman.
Tracz, S. M. (1992). The interpretation of beta weights in path analysis. Multiple
Linear Regression Viewpoints, 19(1), 7–15.
Wothke, W. (1993). Nonpositive definite matrices in structural equation modeling. In K.
A. Bollen & S. J. Long (Eds.), Testing structural equation models (pp. 256–293).
Newbury Park, CA: Sage.
Wothke, W. (2000). Longitudinal and multi-group modeling with missing data.
In T. D. Little, K. U. Schnabel, & J. Baumert (Eds.), Modeling longitudinal and
multiple group data: Practical issues, applied approaches and specific examples
(pp. 1–24). Mahwah, NJ: Lawrence Erlbaum Associates.

Y102005.indb 53

3/22/10 3:25:28 PM

4
SEM Basics

Key Concepts
Model specification and specification error
Fixed, free, and constrained parameters
Under-, just‑, and over-identified models
Recursive versus nonrecursive models
Indeterminancy
Different methods of estimation
Specification search
In this chapter we introduce the basic building blocks of SEM analyses,
which follow a logical sequence of five steps or processes: model specification, model identification, model estimation, model testing, and model modification. In subsequent chapters, we further illustrate these five steps. These
basic building blocks are absolutely essential to conducting SEM models.

4.1 Model Specification
Model specification involves using all of the available relevant theory, research,
and information to develop a theoretical model. Thus, prior to any data collection or analysis, the researcher specifies a particular model that should be
confirmed using variance–covariance data. In other words, available information is used to decide which variables to include in the theoretical model (which
implicitly also involves which variables not to include in the model) and how
these variables are related. Model specification involves determining every relationship and parameter in the model that is of interest to the researcher. Cooley
(1978) indicated that this was the hardest part of structural equation modeling.
A given model is properly specified when the true population model is
deemed consistent with the implied theoretical model being tested—that is,
55

Y102005.indb 55

3/22/10 3:25:28 PM

A Beginner’s Guide to Structural Equation Modeling

the sample covariance matrix S is sufficiently reproduced by the implied theoretical model. The goal of the applied researcher is, therefore, to determine the
best possible model that generates the sample covariance matrix. The sample
covariance matrix implies some underlying, yet unknown, theoretical model
or structure (known as covariance structure), and the researcher’s goal is to
find the model that most closely fits that covariance structure. Take the simple
example of a two-variable situation involving observed variables X and Y. We
know from prior research that X and Y are highly correlated, but why? What
theoretical relationship is responsible for this correlation? Does X influence Y,
does Y influence X, or does a third variable Z influence both X and Y? There
can be many possible reasons why X and Y are related in a particular fashion.
The researcher needs prior research and theories to choose among plausible
explanations and therefore provide the rationale for specifying a model—that
is, testing an implied theoretical model (model specification).
Ultimately, an applied researcher wants to know the extent to which
the true model that generated the data deviates from the implied theoretical model. If the true model is not consistent with the implied theoretical
model, then the implied theoretical model is misspecified. The difference
between the true model and the implied model may be due to errors of
omission and/or inclusion of any variable or parameter. For example,
an important parameter may have been omitted from the model tested
(model did not indicate that X and Y are related), or an important variable may have been omitted (model did not include an important variable, such as amount of education or training). Likewise, an unimportant
parameter and/or unimportant variable may have been included in the
model, that is, there is an error of inclusion.
The exclusion or inclusion of unimportant variables will produce implied
models that are misspecified. Why should we be concerned about this? The
problem is that a misspecified model may result in biased parameter estimates, in other words, estimates that are systematically different from what
they really are in the true model. This bias is known as specification error. In
the presence of specification error, it is likely that one’s theoretical model
may not fit the data and be deemed statistically unacceptable (see model testing in section 4.4). There are a number of procedures available for the detection of specification error so that a more properly specified model may be
evaluated. The model modification procedures are described in section 4.5.

4.2 Model Identification
In structural equation modeling, it is crucial that the researcher resolve the
identification problem prior to the estimation of parameters. In the identification problem, we ask the following question: On the basis of the sample

Y102005.indb 56

3/22/10 3:25:28 PM

SEM Basics

data contained in the sample covariance matrix S and the theoretical model
implied by the population covariance matrix Σ, can a unique set of parameter estimates be found? For example, the theoretical model might suggest
that X + Y = some value, the data might indicate that X + Y = 10, and yet it
may be that no unique solution for X and Y exists. One solution is that X
= 5 and Y = 5, another is that X = 2 and Y = 8, and so on, because there are
an infinite number of possible solutions for this problem, that is, there is
an indeterminacy or the possibility that the data fits more than one implied
theoretical model equally well. The problem is that there are not enough
constraints on the model and the data to obtain unique estimates of X and
Y. Therefore, if we wish to solve this problem, we need to impose some constraints. One such constraint might be to fix the value of X to 1; then Y would
have to be 9. We have solved the identification problem in this instance by
imposing one constraint. However, except for simplistic models, the solution to the identification problem in structural equation modeling is not so
easy (although algebraically one can typically solve the problem).
Each potential parameter in a model must be specified to be either a free
parameter, a fixed parameter, or a constrained parameter. A free parameter is a parameter that is unknown and therefore needs to be estimated.
A fixed parameter is a parameter that is not free, but is fixed to a specified
value, typically either 0 or 1. A constrained parameter is a parameter that is
unknown, but is constrained to equal one or more other parameters.
Model identification depends on the designation of parameters as fixed,
free, or constrained. Once the model is specified and the parameter specifications are indicated, the parameters are combined to form one and
only one Σ (model implied variance–covariance matrix). The problem still
exists, however, in that there may be several sets of parameter values that
can form the same Σ. If two or more sets of parameter values generate
the same Σ, then they are equivalent, that is, yield equivalent models (Lee
& Hershberger, 1990; MacCallum, Wegener, Uchino, & Fabrigar, 1993;
Raykov & Penev, 2001). If a parameter has the same value in all equivalent
sets, then the parameter is identified. If all of the parameters of a model
are identified, then the entire model is identified. If one or more of the
parameters are not identified, then the entire model is not identified.
Traditionally, there have been three levels of model identification. They
depend on the amount of information in the sample variance–covariance
matrix S necessary for uniquely estimating the parameters of the model.
The three levels of model identification are as follows:
1. A model is under-identified (or not identified) if one or more
parameters may not be uniquely determined because there is not
enough information in the matrix S.
2. A model is just‑identified if all of the parameters are uniquely determined because there is just enough information in the matrix S.

Y102005.indb 57

3/22/10 3:25:28 PM

A Beginner’s Guide to Structural Equation Modeling

3. A model is over-identified when there is more than one way of estimating a parameter (or parameters) because there is more than
enough information in the matrix S.
If a model is either just‑ or over-identified, then the model is identified.
If a model is under-identified, then the parameter estimates are not to be
trusted, that is, the degrees of freedom for the model is negative. However,
such a model may become identified if additional constraints are imposed,
that is, the degrees of freedom equal 0 or greater than 0 (positive value).
There are several conditions for establishing the identification of a model.
A necessary, but not the only sufficient condition for identification is the
order condition, under which the number of free parameters to be estimated
must be less than or equal to the number of distinct values in the matrix S,
that is, only the diagonal variances and one set of off‑diagonal covariance
terms are counted. For example, because s12 = s21 in the off-diagonal of the
matrix, only one of these covariance terms is counted. The number of distinct values in the matrix S is equal to p(p + 1)/2, where p is the number of
observed variables. The number of free parameters (saturated model—all
paths) with the number of means = p is equal to p(p + 1)/2 + p = p(p + 3)/2
free parameters. For a sample matrix S with 3 observed variables, there are
six distinct values [3(3 + 1)/2 = 6] and 9 free (independent) parameters
[3(3 + 3)/2] that can be estimated. Consequently, the number of free parameters estimated in any theoretical implied model must be less than or equal
to the number of distinct values in the S matrix. However, this is only one
necessary condition for model identification; it does not by itself imply that
the model is identified. For example, if the sample size is small (n = 10)
relative to the number of variables (p = 20), then not enough information is
available to estimate parameters in a saturated model. This explanation of
the order condition is referred to as the “t rule” by Bollen (1989).
Whereas the order condition is easy to assess, other sufficient conditions are not—for example, the rank condition. The rank condition requires
an algebraic determination of whether each parameter in the model
can be estimated from the covariance matrix S. Unfortunately, proof of
this rank condition is often problematic in practice, particularly for the
applied researcher. However, there are some procedures that the applied
researcher can use. For a more detailed discussion on the rank condition,
we refer to Bollen (1989) or Jöreskog and Sörbom (1988). The basic concepts
and a set of procedures to handle problems in model identification are
discussed next and in subsequent chapters.
Three different methods for avoiding identification problems are
available. The first method is in the measurement model, where we
decide which observed variables measure each latent variable. Either
one indicator for each latent variable must have a factor loading fixed to
1, or the variance of each latent variable must be fixed to 1. The reason

Y102005.indb 58

3/22/10 3:25:29 PM

SEM Basics

for imposing these constraints is to set the measurement scale for each
latent variable, primarily because of indeterminacy between the variance
of the latent variable and the loadings of the observed variables on that
latent variable. Utilizing either of these methods will eliminate the scale
indeterminacy problem, but not necessarily the identification problem,
and so additional constraints may be necessary.
The second method comes into play where reciprocal or nonrecursive
structural models are used; such models are sometimes a source of the
identification problem. A structural model is recursive when all of the
structural relationships are unidirectional (two latent variables are not
reciprocally related), that is, no feedback loops exist whereby a latent variable feeds back upon itself. Nonrecursive structural models include a reciprocal or bidirectional relationship, so that there is feedback—for example,
models that allow product attitude and product interest to influence one
another. For a nonrecursive model, ordinary least squares (OLS; see model
estimation in section 4.3) is not an appropriate method of estimation.
The third method is to begin with a parsimonious (simple) model with a
minimum number of parameters. The model should only include variables
(parameters) considered to be absolutely crucial. If this model is identified,
then you can consider including other parameters in subsequent models.
A second set of procedures involves methods for checking on the identification of a model. One method is Wald’s (1950) rank test. A second,
related method is described by Wiley (1973), Keesling (1972), and Jöreskog
and Sörbom (1988). This test has to do with the inverse of the information matrix and is computed in LISREL. Unfortunately, these methods are
not 100% reliable, and there is no general “necessary and sufficient” test
available for the applied researcher to use. Our advice is to use whatever
methods are available for identification. If you still suspect that there is
an identification problem, follow the recommendation of Jöreskog and
Sörbom (1988). The first step is to analyze the sample covariance matrix
S and save the estimated population matrix Σ. The second step is to analyze the estimated population matrix Σ. If the model is identified, then the
estimates from both analyses should be identical. Another option, often
recommended, is to use different starting values in separate analyses. If
the model is identified, then the estimates should be identical.

4.3 Model Estimation
In this section we examine different methods for estimating parameters
in a model—that is, estimates of the population parameters in a structural
equation model. We want to obtain estimates for each of the parameters
specified in the model that produce the implied matrix Σ, such that the

Y102005.indb 59

3/22/10 3:25:29 PM

A Beginner’s Guide to Structural Equation Modeling

parameter values yield a matrix as close as possible to S, our sample covariance matrix of the observed or indicator variables. When elements in the
matrix S minus the elements in the matrix Σ equal zero (S – Σ = 0), then
χ2 = 0,—that is, one has a perfect model fit to the data.
The estimation process involves the use of a particular fitting function to
minimize the difference between Σ and S. Several fitting functions or estimation procedures are available. Some of the earlier estimation methods
included unweighted or ordinary least squares (ULS or OLS), generalized
least squares (GLS), and maximum likelihood (ML).
The ULS estimates are consistent, have no distributional assumptions
or associated statistical tests, and are scale dependent—that is, changes
in observed variable scale yield different solutions or sets of estimates. In
fact, of all the estimators described here, only the ULS estimation method
is scale dependent. The GLS and ML methods are scale free, which means
that if we transform the scale of one or more of our observed variables,
the untransformed and transformed variables will yield estimates that
are properly related—that is, that differ by the transformation. The GLS
procedure involves a weight matrix W, such as S−1, the inverse of the
sample covariance matrix. Both GLS and ML estimation methods have
desirable asymptotic properties—that is, large sample properties, such as
minimum variance and unbiasedness. Also, both GLS and ML estimation
methods assume multivariate normality of the observed variables (the
sufficient conditions are that the observations are independent and identically distributed and that kurtosis is zero). The weighted-least squares
(WLS) estimation method generally requires a large sample size and as
a result is considered an asymptotically distribution-free (ADF) estimator, which does not depend on the normality assumption. Raykov and
Widaman (1995) further discussed the use of ADF estimators.
If standardization of the latent variables is desired, one may obtain a
standardized solution (and thereby standardized estimates), where the
variances of the latent variables are fixed at 1 by adding the command
line LISREL OUTPUT SS SC to the SIMPLIS program. A separate but
related issue is standardization of the observed variables. When the unit
of measurement for the indicator variables is of no particular interest to
the researcher—that is, arbitrary or irrelevant—then only an analysis of
the correlation matrix is typically of interest. The analysis of correlations
usually gives correct chi-square goodness-of-fit values but estimates the
standard errors incorrectly. There are ways to specify a model, analyze a
correlation matrix, and obtain correct standard errors. For example, the
SEPATH structural equation modeling program by Steiger (1995) does
permit correlation matrix input and computes the correct standard errors.
Since the correlation matrix involves a standardized scaling among
the observed variables, the parameters estimated for the measurement
model—for example, the factor loadings—will be of the same order of

Y102005.indb 60

3/22/10 3:25:29 PM

SEM Basics

magnitude, that is, on the same scale. When the same indicator variables
are measured either over time (longitudinal analysis), for multiple samples,
or when equality constraints are imposed on two or more parameters, an
analysis of the covariance matrix is appropriate and recommended so as
to capitalize on the metric similarities of the variables (Lomax, 1982).
More recently, other estimation procedures have been developed for
the analysis of covariance structure models. Beginning with LISREL,
automatic starting values have been provided for all of the parameter
estimates. These are referred to as initial estimates and involve a fast, noniterative procedure (unlike other methods such as ML, which is iterative).
The initial estimates involve the instrumental variables and least-squares
methods (ULS and two‑stage least-squares method TSLS) developed by
Hagglund (1982). Often, the user may wish to obtain only the initial estimates (for cost efficiency) or to use them as starting values in subsequent
analyses. The initial estimates are consistent and rather efficient relative
to the ML estimator, and have been shown, as in the case of the centroid
method, to be considerably faster, especially in large-scale measurement
models (Gerbing & Hamilton, 1994).
If one can assume multivariate normality of the observed variables,
then moments beyond the second—that is, skewness and kurtosis—can
be ignored. When the normality assumption is violated, parameter estimates and standard errors are suspect. One alternative is to use GLS,
which assumes multivariate normality and stipulates that kurtosis be
zero (Browne, 1974). Browne (1982, 1984) later recognized that the weight
matrix of GLS may be modified to yield ADF or WLS estimates, standard
errors, and test statistics. Others (Bentler, 1983; Shapiro, 1983) have developed more general classes of ADF estimators. All of these methods are
based on the GLS method and specify that the weight matrix be of a certain form, although none of these methods takes multivariate kurtosis into
account. Research by Browne (1984) suggests that goodness‑of‑fit indices
and standard errors of parameter estimates derived under the assumption of multivariate normality should not be employed if the distribution
of the observed variables has a nonzero value for kurtosis.
An implicit assumption of ML estimators is that information contained
in the first and second order moments (mean, and variance, respectively)
of the observed variables is sufficient so that information contained in
higher-order moments (skewness and kurtosis) can be ignored. If the
observed variables are interval scaled and multivariate normal, then
the ML estimates, standard errors, and chi-square test are appropriate.
However, if the observed variables are ordinal-scaled and/or extremely
skewed or peaked (nonnormally distributed), then the ML estimates,
standard errors, and chi-square test may not be robust.
The use of binary and ordinal response variables in structural equation modeling was pioneered by Muthén (1982, 1984). Muthén proposed

Y102005.indb 61

3/22/10 3:25:29 PM

A Beginner’s Guide to Structural Equation Modeling

a three‑stage limited-information, GLS estimator that provided a large
sample chi-square test of the model and large sample standard errors. The
Muthén categorical variable methodology (CVM) is believed to produce
more suitable coefficients of association than the ordinary Pearson product moment correlations and covariance applied to ordered categorical
variables (Muthén, 1983). This is especially the case with markedly skewed
categorical variables, where correlations must be adjusted to assume values throughout the −1 to +1 range, as is done in the PRELIS program.
The PRELIS computer program handles ordinal variables by computing a polychoric correlation for two ordinal variables (Olsson, 1979), and
a polyserial correlation for an ordinal and an interval variable (Olsson,
Drasgow, & Dorans, 1982), where the ordinal variables are assumed to have
an underlying bivariate normal distribution, which is not necessary with
the Muthén approach. All correlations (Pearson, polychoric, and polyserial) are then used by PRELIS to create an asymptotic covariance matrix for
input into LISREL. The reader is cautioned to not directly use mixed types
of correlation matrices or covariance matrices in a LISREL–SIMPLIS program, but instead use an asymptotic variance–covariance matrix produced
by PRELIS along with the sample variance–covariance matrix as input in
a LISREL–SIMPLIS or LISREL matrix program. The Satorra–Bentler scaled
chi-square would then be reported for the robust model-fit measure.
During the past 15 or 20 years, we have seen considerable research on
the behavior of methods of estimation under various conditions. The most
crucial conditions are characterized by a lack of multivariate normality
and interval level variables. When the data are generated from nonnormally distributed populations and/or represent discrete variables, the
normal theory estimators of standard errors and model-fit indices discussed in chapter 5 could be suspect. However, recent simulation research
by Lei and Lomax (2005) indicated that the ML and GLS estimators are
quite comparable in the case of small to moderate nonnormality for interval data (bias is generally quite small and, in fact, ML tends to slightly
outperform GLS). Similar results were obtained by Fan & Wang (1998).
In the case of severe nonnormality for interval data, one of the distribution free or weighted procedures (ADF, WLS, or GLS) is recommended
(Lomax, 1989). In dealing with noninterval variables, the research indicates that only when categorical data show small to moderate skewness
and kurtosis values (range of −1 to +1, or −1.5 to +1.5) should ML be used.
When these conditions are not met, several options already mentioned are
recommended. These include the use of tetrachoric, polyserial, and polychoric correlations rather than Pearson product‑moment correlations, or
the use of distribution‑free or weighted procedures available in the SEM
software. Considerable research remains to be conducted to determine
what the optimal estimation procedure is for a given set of conditions. In
summary, we recommend the use of ML estimation for slight to moderate

Y102005.indb 62

3/22/10 3:25:29 PM

SEM Basics

nonnormal interval and ordinal data, and ADF, WLS, or GLS estimation
for severely nonnormal interval and ordinal data.

4.4 Model Testing
Once the parameter estimates are obtained for a specified SEM model,
the researcher should determine how well the data fit the model. In other
words, to what extent is the theoretical model supported by the obtained
sample data? There are two ways to think about model fit. The first is to
consider some global type omnibus test of the fit of the entire model. The
second is to examine the fit of individual parameters in the model.
We first consider the global tests in SEM known as model-fit criteria.
Unlike many statistical procedures that have a single, most powerful fit
index—for example, F test in ANOVA—in SEM there are an increasingly
large number of model-fit indices. Many of these measures are based on a
comparison of the model implied covariance matrix Σ to the sample covariance matrix S. If Σ and S are similar in some fashion, then one may say
that the data fit the theoretical model. If Σ and S are quite different, then
one may say that the data do not fit the theoretical model. We explain
model-fit indices in more detail in chapter 5.
Second, we consider the individual parameters of the model. Three
main features of the individual parameters can be considered. One feature is whether a free parameter is significantly different from zero. Once
parameter estimates are obtained, standard errors for each estimate are
also computed. A ratio of the parameter estimate to the estimated standard error can be formed as a critical value, which is assumed to be normally distributed (unit normal distribution)—that is, the critical value
equals the parameter estimate divided by the standard error of the parameter estimate. If the critical value exceeds the expected value at a specified
a level—for example, a = .05, two tailed test, tabled t = 1.96—then that
parameter is significantly different from zero. The parameter estimate,
standard error, and critical value are routinely provided in the computer
output for a model. A second feature is whether the sign of the parameter
agrees with what is expected from the theoretical model. For example, if
the expectation is that more education will yield a higher income level,
then an estimate with a positive sign would support that expectation.
A third feature is that parameter estimates should make sense—that is,
they should be within an expected range of values. For instance, variances should not have negative values and correlations should not exceed
1. Thus, all free parameters should be in the expected direction, be statistically different from zero, and be meaningfully interpreted.

Y102005.indb 63

3/22/10 3:25:29 PM

A Beginner’s Guide to Structural Equation Modeling

4.5 Model Modification
If the fit of the implied theoretical model is not as strong as one would
like (which is typically the case with an initial model), then the next
step is to modify the model and subsequently evaluate the new modified model. In order to determine how to modify the model, there are a
number of procedures available for the detection of specification errors
so that more properly specified alternative models may be evaluated during respecification process. In general, these procedures are used for performing what is called a specification search (Leamer, 1978). The purpose
of a specification search is to alter the original model in the search for
a model that is better fitting in some sense and yields parameters having practical significance and substantive meaning. If a parameter has
no substantive meaning to the applied researcher, then it should never
be included in a model. Substantive interest must be the guiding force
in a specification search; otherwise, the resultant model will not have
practical value or importance. There are procedures designed to detect
and correct for specification errors. Typically, applications of structural
equation modeling include some type of specification search, informal or
formal, although the search process may not always be explicitly stated
in a research report.
An obvious intuitive method is to consider the statistical significance of
each parameter estimated in the model. One specification strategy would
be to fix parameters that are not statistically significant—that is, have small
critical values, to 0 in a subsequent model. Care should be taken, however,
because statistical significance is related to power and sample size (see
chapter 5); parameters may not be significant with small samples but significant with larger samples. Also, substantive theoretical interests must
be considered. If a parameter is not significant, but is of sufficient substantive interest, then the parameter should probably remain in the model.
The guiding rule should be that the parameter estimates make sense to
you. If an estimate makes no sense to you, how are you going to explain it,
how is it going to be of substantive value or meaningful?
Another intuitive method of examining misspecification is to examine
the residual matrix, that is, the differences between the observed covariance matrix S and the model-implied covariance matrix Σ; these are
referred to as fitted residuals in the LISREL program output. These values
should be small in magnitude and should not be larger for one variable
than another. Large values overall indicate serious general model misspecification, whereas large values for a single variable indicate misspecification for that variable only, probably in the structural model (Bentler, 1989).
Standardized or normalized residuals can also be examined. Theoretically,
these can be treated like standardized z scores, and hence problems can

Y102005.indb 64

3/22/10 3:25:29 PM

SEM Basics

be more easily detected from the standardized residual matrix than from
the unstandardized residual matrix. Large standardized residuals (larger
than, say, 1.96 or 2.58) indicate that a particular covariance structure is not
well explained by the model. The model should be examined to determine
ways in which this particular covariance structure could be explained, for
example, by freeing some parameters in the model.
Sörbom (1975) considered misspecification of correlated measurement
error terms in the analysis of longitudinal data. Sörbom proposed considering the first order partial derivatives, which have values of zero for free
parameters and nonzero values for fixed parameters. The largest value,
in absolute terms, indicates the fixed parameter most likely to improve
model fit. A second model, with this parameter now free, is then estimated and goodness of fit assessed. Sörbom defines an acceptable fit as
occurring when the difference between the two model chi-square values
is not significant. The derivatives of the second model are examined, and
the process continues until an acceptable fit is achieved. This procedure,
however, is restricted to the derivatives of the observed variables and provides indications of misspecification only in terms of correlated measurement error.
More recently, other procedures have been developed to examine model
specification. In the LISREL–SIMPLIS program, modification indices are
reported for all nonfree parameters. These indices were developed by
Sörbom (1986) and represent an improvement over the first order partial
derivatives already described. A modification index for a particular nonfree parameter indicates that if this parameter were allowed to become
free in a subsequent model, then the chi-square goodness-of-fit value
would be predicted to decrease by at least the value of the modification
index. In other words, if the value of the modification index for a nonfree
parameter is 50, then when this parameter is allowed to be free in a subsequent model, the value of chi-square will decrease by at least 50. Thus,
modification indices would suggest ways that the model might be altered
by allowing the corresponding parameters to become free to be estimated
with the researcher arriving at a better fitting model. As reported in an
earlier LISREL manual (Jöreskog & Sörbom, 1988), “This procedure seems
to work well in practice” (p. 44).
The LISREL program also provides squared multiple correlations for the
observed variables in the measurement equations. These values indicate
how well the observed variables serve as measures of the latent variables
(reliability measure) and are scaled from 0 to 1. Squared multiple correlations are also given for the variables in the structural equations. These
values serve as an indication of the strength of the structural relationships
(prediction measure) and are also scaled from 0 to 1.
A relatively new index, the expected parameter change, now appears in
the LISREL program computer output. The expected parameter change

Y102005.indb 65

3/22/10 3:25:30 PM

A Beginner’s Guide to Structural Equation Modeling

(EPC) statistic in the LISREL program computer output indicates the estimated change in the magnitude and direction of each nonfree parameter
if set free to be estimated (rather than the predicted change in the goodness‑of‑fit test as with the modification indices). This could be useful, for
example, if the sign of the potential free parameter is not in the expected
direction (positive instead of negative). This would suggest that such a
parameter should remain fixed.
Empirical research suggests that specification searches are most successful when the model tested is very similar to the model that generated
the data. More specifically, these studies begin with a known true model
from which sample data are generated. The true model is then intentionally misspecified. The goal of the specification search is to begin with the
misspecified model and determine whether the true model can be located
as a result of the search. If the misspecified model is more than two or
three parameters different from the true model, then it is difficult to locate
the true model. Unfortunately, in these studies the true model was almost
never located through the specification search, regardless of the search
procedure or combination of procedures that were used (Gallini, 1983;
Gallini & Mandeville, 1984; Saris & Stronkhorst, 1984; MacCallum, 1986;
Baldwin & Lomax, 1990; Tippets, 1992).
What is clear is that there is no single existing procedure sufficient for
finding a properly specified model. As a result, there has been a flurry
of research in recent years to determine what combination of procedures
is most likely to yield a properly specified model (Chou & Bentler, 1990;
Herbing & Costner, 1985; Kaplan, 1988, 1989, 1990; MacCallum, 1986;
Saris, Satorra & Sörbom, 1987; Satorra & Saris, 1985; Silvia & MacCallum,
1988). No optimal strategy has been found. A computer program known as
TETRAD was developed by Glymour, Scheines, Spirtes, and Kelly (1987),
and the new version, TETRAD II (Spirtes, Scheines, Meek, & Glymour,
1994), thoughtfully reviewed by Wood (1995), offers new search procedures. A newer specification search procedure, known as Tabu, recently
developed by Marcoulides, Drezner, and Schumacker (1998) can today
readily provide a set of optimum models. If one selected all of the paths in
the model as optional, then all possible models would be listed; for example, a multiple regression equation with 17 independent variables and 1
dependent variable would yield 217 or 131,072 regression models, not all of
which would be theoretically meaningful. Selection of the “best” equation
would require the use of some fit criteria for comparing models. Applying
Tabu in SEM, for example, χ2 – df, AIC, or BIC would be used for selecting
best models. Current modeling software permits the formulation of all
possible models; however, the outcome of any specification search should
still be guided by theory and practical considerations as well as the time
and cost of acquiring the data.

Y102005.indb 66

3/22/10 3:25:30 PM

SEM Basics

Given our lengthy discussion about specification search procedures,
some practical advice is warranted for the researcher. The following is
our suggested eight-step procedure for a specification search:
1. Let substantive theory and prior research guide your model
specification.
2. When you are satisfied that Rule 1 has been met, test your implied
theoretical model and move to Rule 3.
3. Conduct a specification search, first on the measurement model,
and then on the structural model.
4. For each model tested, look to see if the parameters are of the
expected magnitude and direction, and examine several appropriate goodness‑of‑fit indices.
Steps 5 through 7 can be followed in an iterative fashion. For
example, you might go from Step 5 to Step 6, and successively
on to Steps 7, 6, 5, and so on.
5. Examine the statistical significance of the nonfixed parameters.
Look to see if any nonfixed parameters should be fixed in a subsequent model.
6. Examine the modification indices, expected parameter change
statistics. Look to see if any fixed parameters should be freed in a
subsequent model.
7. Consider examining the standardized residual matrix to see if
anything suspicious is occurring (larger values for a particular
observed variable).
8. Once you determine a final acceptable model, cross‑validate it with
a new sample, or use half of the sample to find a properly specified
model and use the other half to check it (cross-validation index, or
CVI), or report a single sample cross-validation index (ECVI) for
alternative models (Cudeck & Browne, 1983; Kroonenberg & Lewis,
1982). Cross-validation procedures are discussed in chapter 12.

4.6 Summary
In this chapter we considered the basics of structural equation modeling.
The chapter began with a look at model specification (fixed, free, and constrained parameters) and then moved on to model identification (under-,
just‑, and over-identified models). Next, we discussed the various types of

Y102005.indb 67

3/22/10 3:25:30 PM

A Beginner’s Guide to Structural Equation Modeling

estimation procedures. Here we considered each estimation method, its
underlying assumptions, and some general guidelines as to when each
is appropriate. We then moved on to a general discussion of model testing, where the fit of a given model is assessed. Finally, we described the
specification search process, where information is used to arrive at a more
properly specified model that is theoretically meaningful. Troubleshooting
tips summarizing these key issues are provided in Box 4.1.

BOX 4.1

Troubleshooting Tips

Issue
Identification
problem

Estimation
method
Specification
search

Suggestions
Solutions include fixing parameters (either latent variable
variances or one factor loading for each latent variable),
avoiding nonrecursive models, utilizing parsimonious
models, or determining if a positive degree of freedom exists
when subtracting total number of elements in matrix from
number of free parameters to be estimated in the model.
For normal and slight to moderate nonnormal interval and
ordinal data, use ML; otherwise consider WLS, ADF, GLS, or
CVM methods.
Examine the statistical significance of free parameters,
standardized residuals, modification indices, goodness-of-fit
indices, squared multiple correlations, as well as expected
parameter change.

In chapter 5, we discuss the numerous goodness-of-fit indices in the
LISREL computer output to determine whether a model is parsimonious, which alternative models are better, and to examine submodels
(nested models). We classify the model-fit indices according to whether
a researcher is testing model fit, seeking a more parsimonious model
(complex to simple), or comparing nested models. In addition, we discuss
hypothesis testing, parameter significance, power, and sample size, as
these affect our interpretation of model fit and statistical significance of
parameter estimates.
Exercises
1. Define model specification.
2. Define model identification.
3. Define model estimation.
4. Define model testing.

Y102005.indb 68

3/22/10 3:25:30 PM

SEM Basics

5. Define model modification.
6. Determine the number of distinct values (variances and covariances) in the following variance–covariance matrix S:


1.0

S = .25


.35


1.0
.45







1.00 


7. How many distinct values are in a variance–covariance matrix
for the following variables {hint: [p(p + 1)/2]}:
a. Five variables
b. Ten variables
8. A saturated model with p variables has p(p + 3)/2 free parameters. Determine the number of free parameters for the following number of variables in a model:
a. Three observed variables
b. Five observed variables
c. Ten observed variables

References
Baldwin, B., & Lomax, R. G. (1990). Measurement model specification error in LISREL
structural equation models. Paper presented at the annual meeting of the
American Educational Research Association, Boston.
Bentler, P. M. (1983). Some contributions to efficient statistics in structural models: Specification and estimation of moment structures. Psychometrika, 48,
493–517.
Bentler, P. M. (1989). Theory and implementation of EQS: A structural equations program. Los Angeles: BMDP Statistical Software.
Bollen, K. A. (1989). Structural equations with latent variables. New York: Wiley.
Browne, M. W. (1974). Generalized least‑squares estimators in the analysis of covariance structures. South African Statistical Journal, 8, 1–24.
Browne, M. W. (1982). Covariance structures. In D. M. Hawkins (Ed.), Topics in applied
multivariate analysis (pp. 72–141). Cambridge: Cambridge University Press.
Browne, M. W. (1984). Asymptotically distribution‑free methods for the analysis of
covariance structures. British Journal of Mathematical and Statistical Psychology,
37, 62–83.

Y102005.indb 69

3/22/10 3:25:30 PM

A Beginner’s Guide to Structural Equation Modeling

Chou, C. ‑P., & Bentler, P. M. (1990). Power of the likelihood ratio, Lagrange multiplier, and Wald tests for model modification in covariance structure analysis. Paper
presented at the annual meeting of the American Educational Research
Association, Boston.
Cooley, W. W. (1978). Explanatory observational studies. Educational Researcher,
7(9), 9–15.
Cudeck, R., & Browne, M. W. (1983). Cross‑validation of covariance structures.
Multivariate Behavioral Research, 18, 147–167.
Fan, X., & Wang, L. (1998). Effects of potential confounding factors on fit indices and parameter estimates for true and misspecified models. Structural
Equation Modeling: A Multidisciplinary Journal, 5, 701–735.
Gallini, J. K. (1983). Misspecifications that can result in path analysis structures.
Applied Psychological Measurement, 7, 125–137.
Gallini, J. K., & Mandeville, G. K. (1984). An investigation of the effect of sample
size and specification error on the fit of structural equation models. Journal of
Experimental Education, 53, 9–19.
Gerbing, D. W., & Hamilton, J. G. (1994). The surprising viability of a simple alternate estimation procedure for construction of large-scale structural equation measurement models. Structural Equation Modeling: A Multidisciplinary
Journal, 1, 103–115.
Glymour, C. R., Scheines, R., Spirtes, P., & Kelly, K. (1987). Discovering causal structure. Orlando: Academic.
Hagglund, G. (1982). Factor analysis by instrumental variable methods.
Psychometrika, 47, 209–222.
Herbing, J. R., & Costner, H. L. (1985). Respecification in multiple indicator models. In H. M. Blalock, Jr. (Ed.), Causal models in the social sciences (2nd ed.,
pp. 321–393). New York: Aldine.
Jöreskog, K. G., & Sörbom, D. (1988). LISREL 7: A guide to the program and applications. Chicago: SPSS.
Kaplan, D. (1988). The impact of specification error on the estimation, testing, and
improvement of structural equation models. Multivariate Behavioral Research,
23, 69–86.
Kaplan, D. (1989). Model modification in covariance structure analysis: Application
of the parameter change statistic. Multivariate Behavioral Research, 24,
285–305.
Kaplan, D. (1990). Evaluating and modifying covariance structure models: A
review and recommendation. Multivariate Behavioral Research, 25, 137–155.
Keesling, J. W. (1972). Maximum likelihood approaches to causal flow analysis.
Unpublished dissertation. University of Chicago, Department of Education.
Kroonenberg, P. M., & Lewis, C. (1982). Methodological issues in the search for
a factor model: Exploration through confirmation. Journal of Educational
Statistics, 7, 69–89.
Leamer, E. E. (1978). Specification searches. New York: Wiley.
Lee, S., & Hershberger, S. (1990). A simple rule for generating equivalent models in
covariance structure modeling. Multivariate Behavioral Research, 25, 313–334.
Lei, M., & Lomax, R. G. (2005). The effect of varying degrees of nonnormality in
structural equation modeling. Structural Equation Modeling: A Multidisciplinary
Journal, 12, 1–27.

Y102005.indb 70

3/22/10 3:25:30 PM

SEM Basics

Lomax, R. G. (1982). A guide to LISREL‑type structural equation modeling. Behavior
Research Methods & Instrumentation, 14, 1–8.
Lomax, R. G. (1989). Covariance structure analysis: Extensions and developments.
In B. Thompson (Ed.), Advances in social science methodology (Vol. 1, pp. 171–204).
Greenwich, CT: JAI.
MacCallum, R. C. (1986). Specification searches in covariance structure modeling.
Psychological Bulletin, 100, 107–120.
MacCallum, R. C., Wegener, D. T., Uchino, B. N., & Fabrigar, L. R. (1993). The problem of equivalent models in applications of covariance structure analysis.
Psychological Bulletin, 114, 185–199.
Marcoulides, G. A., Drezner, Z., & Schumacker, R. E. (1998). Model specification searches in structural equation modeling using Tabu search. Structural
Equation Modeling: A Multidisciplinary Journal, 5, 365–376.
Muthén, B. (1982). Some categorical response models with continuous latent variables. In K. G. Jöreskog & H. Wold (Eds.), Systems under indirect observation:
Causality, structure, prediction, Part I (pp. 65–79). Amsterdam: North‑Holland.
Muthén, B. (1983). Latent variable structural equation modeling with categorical
data. Journal of Econometrics, 22, 43–65.
Muthén, B. (1984). A general structural equation model with dichotomous, ordered
categorical, and continuous latent variable indicators. Psychometrika, 49,
115–132.
Olsson, U. (1979). Maximum likelihood estimation of the polychoric correlation
coefficient. Psychometrika, 44, 443–460.
Olsson, U., Drasgow, F., & Dorans, N. J. (1982). The polyserial correlation coefficient. Psychometrika, 47, 337–347.
Raykov, T., & Penev, S. (2001). The problem of equivalent structural equation
models: An individual residual perspective. In G. A. Marcoulides & R. E.
Schumacker (Eds.), New developments and techniques in structural equation modeling (pp. 297–321). Mahwah, NJ: Lawrence Erlbaum.
Raykov, T., & Widaman, K. F. (1995). Issues in applied structural equation modeling research. Structural Equation Modeling: A Multidisciplinary Journal, 2,
289–318.
Saris, W. E., Satorra, A., & Sörbom, D. (1987). The detection and correction of
specification errors in structural equation models. In C. C. Clogg (Ed.),
Sociological methodology (pp. 105–130). Washington, DC: American Sociological
Association.
Saris, W. E., & Stronkhorst, L. H. (1984). Causal modeling in nonexperimental
research: An introduction to the LISREL approach. Amsterdam: Sociometric
Research Foundation.
Satorra, A., & Saris, W. E. (1985). Power of the likelihood ratio test in covariance
structure analysis. Psychometrika, 50, 83–90.
Shapiro, A. (1983). Asymptotic distribution theory in the analysis of covariance
structures (a unified approach). South African Statistical Journal, 17, 33–81.
Silvia, E. S. M., & MacCallum, R. (1988). Some factors affecting the success of specification searches in covariance structure modeling. Multivariate Behavioral
Research, 23, 297–326.
Sörbom, D. (1975). Detection of correlated errors in longitudinal data. British
Journal of Mathematical and Statistical Psychology, 27, 229–239.

Y102005.indb 71

3/22/10 3:25:30 PM

A Beginner’s Guide to Structural Equation Modeling

Sörbom, D. (1986). Model modification (Research Report 86‑3). University of Uppsala,
Department of Statistics, Uppsala, Sweden.
Spirtes, P., Scheines, R., Meek, C., & Glymour, C. (1994). TETRAD II: Tools for causal
modeling. Hillsdale, NJ: Lawrence Erlbaum.
Steiger, J. H. (1995). SEPATH. In STATISTICA 5.0. Tulsa, OK: StatSoft.
Tippets, E. (1992). A comparison of methods for evaluating and modifying covariance
structure models. Paper presented at the annual meeting of the American
Educational Research Association, San Francisco.
Wald, A. (1950). A note on the identification of economic relations. In T. C.
Koopmans (Ed.), Statistical inference in dynamic economic models (pp. 238–244).
New York: Wiley.
Wiley, D. E. (1973). The identification problem for structural equation models with
unmeasured variables. In A. S. Goldberger & O. D. Duncan (Eds.), Structural
equation models in the social sciences (pp. 69–83). New York: Seminar.
Wood, P. K. (1995). Toward a more critical examination of structural equation models. Structural Equation Modeling: A Multidisciplinary Journal, 2, 277–287.

Y102005.indb 72

3/22/10 3:25:31 PM

5
Model Fit

Key Concepts
Confirmatory models, alternative models, model generating
Specification search
Saturated models and independence models
Model fit, model comparison, and model parsimony fit indices
Measurement model versus structural model interpretation
Model and parameter significance
Power and sample size determination
In chapter 4, we considered the basic building blocks of SEM, namely,
model specification, model identification, model estimation, model testing,
and model modification. These five steps fall into three main approaches
for going from theory to a SEM model in which the covariance structure
among variables is analyzed. In the confirmatory approach, a researcher
hypothesizes a specific theoretical model, gathers data, and then tests
whether the data fit the model. In this approach, the theoretical model is
either confirmed or disconfirmed, based on a chi-square statistical test of
significance and/or meeting acceptable model-fit criteria. In the second
approach using alternative models, the researcher creates a limited number of theoretically different models to determine which model the data
fit best. When these models use the same data set, they are referred to as
nested models. The alternative approach conducts a chi-square difference
test to compare each of the alternative models. The third approach, model
generating, specifies an initial model (theoretical model), but usually the
data do not fit this initial model at an acceptable model-fit criterion level, so
modification indices are used to add or delete paths in the model to arrive
at a final best model. The goal in model generating is to find a model that
the data fit well statistically, but that also has practical and substantive
theoretical meaning. The process of finding the best-fitting model is also
referred to as a specification search, implying that if an initially specified
73

Y102005.indb 73

3/22/10 3:25:31 PM

A Beginner’s Guide to Structural Equation Modeling

model does not fit the data, then the model is modified in an effort to
improve the fit (Marcoulides & Drezner, 2001; 2003). Recent advances in
Tabu search algorithms have permitted the generation of a set of models that
the data fit equally well with a final determination by the researcher of
which model to accept (Marcoulides, Drezner, & Schumacker, 1998).

5.1 Types of Model-Fit Criteria
Finding a statistically significant theoretical model that also has practical
and substantive meaning is the primary goal of using structural equation
modeling to test theories. A researcher typically uses the following three
criteria in judging the statistical significance and substantive meaning of
a theoretical model:
1. The first criterion is the nonstatistical significance of the chi-square
test and the root-mean-square error of approximation (RMSEA)
values, which are global fit measures. A nonstatistically significant
chi-square value indicates that the sample covariance matrix and
the reproduced model implied covariance matrix are similar. A
RMSEA value less than or equal to .05 is considered acceptable.
2. The second criterion is the statistical significance of individual
parameter estimates for the paths in the model, which are values
computed by dividing the parameter estimates by their respective
standard errors. This is referred to as a t value, and is typically
compared to a tabled t value of 1.96 at the .05 level of significance
(two-tailed). [Note: LISREL 8.8 student version now reports the
standard error, z-value, and p-value for each parameter.]
3. The third criterion is the magnitude and direction of the parameter estimates, paying particular attention to whether a positive
or negative coefficient makes sense for the parameter estimate.
For example, it would not be theoretically meaningful to have a
negative parameter (coefficient) relating number of hours spent
studying and grade point average.
We now describe the numerous criteria for assessing model fit, and offer
suggestions on how and when these criteria might be used. Determining
model fit is complicated because several model-fit criteria have been
developed to assist in interpreting structural equation models under different model-building assumptions. In addition, the determination of
model fit in structural equation modeling is not as straightforward as
it is in other statistical approaches in multivariable procedures, such as

Y102005.indb 74

3/22/10 3:25:31 PM

Model Fit

the analysis of variance, multiple regression, discriminant analysis, multivariate analysis of variance, and canonical correlation analysis. These
multivariable methods use observed variables that are assumed to be
measured without error and have statistical tests with known distributions. Many SEM model-fit indices have no single statistical test of significance that identifies a correct model, given the sample data, especially
since equivalent models or alternative models can exist that yield exactly the
same data to model fit.
Chi-square (c2) is the only statistical test of significance for testing the
theoretical model (see Table 5.1 for fit indices and their interpretation). The
chi-square value ranges from zero for a saturated model with all paths
included to a maximum value for the independence model with no paths
included. The theoretical model chi-square value lies somewhere between
these two extremes. This can be visualized as follows:
Saturated model
(all paths in model)
c2 = 0

Independence model
(no paths in model)
c2 = maximum value

A chi-square value of zero indicates a perfect fit or no difference between
values in the sample covariance matrix S and the reproduced implied covariance matrix Σ that was created, based on the specified (implied) theoretical
model. Obviously, a theoretical model in SEM with all paths specified is of
limited interest (saturated model). The goal in structural equation modeling is to achieve a parsimonious model with a few substantive meaningful
paths and a nonsignificant chi-square value close to the saturated model
value of zero, thus indicating little difference between the sample covariance matrix and the reproduced implied covariance matrix. The difference
between these two covariance matrices is output in a residual matrix (add
command line Print Residual to SIMPLIS program). When the chi-square
value is nonsignificant (close to zero), residual values in the residual matrix
are close to zero, indicating that the theoretical implied model fits the sample data, hence there is little difference between the sample covariance
matrix and the model implied (reproduced) covariance matrix.
Many of the model-fit criteria are computed-based on knowledge of
the saturated model, independence model, sample size, degrees of freedom, and/or the chi-square values to formulate an index of model fit
that ranges in value from 0 (no fit) to 1 (perfect fit). These various modelfit indices, however, are subjectively interpreted when determining an
acceptable model fit. Some researchers have suggested that a structural
equation model with a model-fit value of .90 or .95 or higher is acceptable
(Baldwin, 1989; Bentler & Bonett, 1980), whereas more recently a noncentrality parameter close to zero [NCP = max(0, c2 − df )] has been suggested

Y102005.indb 75

3/22/10 3:25:31 PM

A Beginner’s Guide to Structural Equation Modeling

Table 5.1
Model-Fit Criteria and Acceptable Fit Interpretation
Model-Fit Criterion

Acceptable Level

Chi-square

Tabled c2 value

Goodness-of-fit index
(GFI)
Adjusted GFI (AGFI)

0 (no fit) to 1 (perfect fit)

Root-mean square residual
(RMR)
Standardized RMR
(SRMR)
Root-mean-square error of
approximation (RMSEA)
Tucker–Lewis Index (TLI)

Researcher defines level

Normed fit index (NFI)

0 (no fit) to 1 (perfect fit)

Parsimony fit index (PNFI)

0 (no fit) to 1 (perfect fit)

Akaike information
criterion (AIC)

0 (perfect fit) to positive
value (poor fit)

0 (no fit) to 1 (perfect fit)

< .05
.05 to .08
0 (no fit) to 1 (perfect fit)

Interpretation
Compares obtained c2 value
with tabled value for given df
Value close to .90 or .95 reflect
a good fit
Value adjusted for df, with .90
or .95 a good model fit
Indicates the closeness of Σ to
S matrices
Value less than .05 indicates a
good model fit
Value of .05 to .08 indicate
close fit
Value close to .90 or .95 reflects
a good model fit
Value close to .90 or .95 reflects
a good model fit
Compares values in alternative
models
Compares values in
alternative models

(Browne & Cudeck, 1993; Steiger, 1990). The various structural equation
modeling programs report a variety of model-fit criteria, and thus only
those output by LISREL are shown in this chapter. It is recommended that
various model-fit criteria be used in combination to assess model fit, model
comparison, and model parsimony as global fit measures (Hair, Anderson,
Tatham, & Black, 1992).
Some of the fit indices are computed given knowledge of the null model c2
(independence model, where the covariance terms are assumed to be zero
in the model), null model df, hypothesized model c2, hypothesized model
df, number of observed variables in the model, number of free parameters
in the model, and sample size. The formula for the goodness-of-fit index
(GFI), normed fit index (NFI), relative fit index (RFI), incremental fit index
(IFI), Tucker-Lewis index (TLI), comparative fit index (CFI), model AIC,
null AIC, and RMSEA using these values are as follows:
GFI = 1 – [c2model/c2null]
NFI = (c2null − c2model)/c2null
RFI = 1 – [(c2model/dfmodel)/(c2null/dfnull)]
IFI = (c2null − c2model)/(c2null − dfmodel)
TLI = [(c2null/dfnull) − (c2model/dfmodel)]/[(c2null/dfnull) − 1]
CFI = 1 – [(c2model − dfmodel)/(c2null − dfnull)]

Y102005.indb 76

3/22/10 3:25:31 PM

Model Fit

Model AIC = c2model + 2q (number of free parameters)
Null AIC = c2 null + 2q (number of free parameters)
2
RMSEA = [ χ Model
− dfModel ]/[( N − 1)dfModel ]

These model-fit statistics can also be expressed in terms of the noncentrality parameter (NCP), designated by l. The estimate of NCP (l) using
the maximum likelihood chi-square is c2 − df. A simple substitution reexpresses these model-fit statistics using NCP. For example, CFI, TLI, and
RMSEA are as follows:
CFI = 1 – [ lModel/lNull]
TLI = 1 − [(lModel/df Model)/(lNull/df Null)]
RMSEA = λModel /[( N − 1)dfModel ]
Bollen and Long (1993), as well as Hu and Bentler (1995), have thoroughly
discussed several issues related to model fit, and we recommend reading
their assessments of how model-fit indices are affected by small sample bias,
estimation methods, violation of normality and independence, and model
complexity, and for an overall discussion of the various model-fit indices.
5.1.1 LISREL–SIMPLIS Example
Our purpose in this chapter is to better understand the model-fit criteria output by LISREL–SIMPLIS. The theoretical model in Figure 5.1a is
analyzed to aid in the understanding of model-fit criteria, significance
of parameter estimates, and power and sample size determination. The
theoretical basis for this model is discussed in more detail in chapter 8.
The two factor confirmatory model is based on data from Holzinger and
Swineford (1939) using data collected on 26 psychological tests from 301
children in a suburban school district of Chicago. Over the years, different
subsamples of the children and different subsets of the variables in this
dataset have been analyzed and presented in various multivariate statistics textbooks (Gorsuch, 1983; Harmon, 1976), and SEM software program
guide (Jöreskog & Sörbom, 1993, example 5, pp. 2–28). For our analysis, we
used data on the first six psychological variables for all 301 subjects. The
theoretical model is depicted in Figure 5.1a.
5.1.1.1 Data
The LISREL program can easily import many different file types. To
import the SPSS data file holz.sav, simply click on File, then select Import
Data. Next select SPSS for Windows(*.sav) from the pull-down menu for
Files of type: and then select HOLZ data file. (Note: The data file may be
in a different location, so you may have to search to locate it).

Y102005.indb 77

3/22/10 3:25:32 PM

A Beginner’s Guide to Structural Equation Modeling

0.39

visperc

0.78
0.81

cubes

0.4

Spatial

0.57
0.68

lozenges
0.46

0.27

parcomp

0.85

Verbal

0.85
0.27

sencomp

0.30

wordmean

0.84

Figure 5.1a
Common factor model. (From Holzinger, K. J., & Swineford, F. A. [1939]. A study in factor
analysis: The stability of a bi-factor solution. Supplementary Educational Monographs, No. 48.
Chicago: University of Chicago, Dept. of Education.)

Y102005.indb 78

3/22/10 3:25:33 PM

Model Fit

After clicking on Open, a Save As dialog box appears to save a PRELIS
System File, so enter holz.psf.

A spreadsheet should appear that contains the variable names and data.
Also, an expanded tool bar menu appears that begins with File, includes
Edit, Data, Transformation, Statistics, Graphs, etc., and ends with the Help
command. The File command also permits the use of an Export LISREL
Data option. The File, then Import Data option should be used to save a
PRELIS System File whenever possible to take advantage of data screening, imputing missing values, computation of normal scores, output data
options, and many other features in LISREL–PRELIS. For our purposes,
click on Statistics, then select the Output Options. The Output dialog box
will be used to save a correlation matrix file (holz.cor), a means file (holz.
me), and standard deviations file (holz.sd) for the variables we will use to
analyze our theoretical model in Figure 5.1a. The correlation, means, and
standard deviation files must be saved (or moved) to the same directory
as the LISREL–SIMPLIS program file. Click OK and descriptive statistics
appear in the computer output (frequencies, means, standard deviations,
skewness, kurtosis, etc.).

Y102005.indb 79

3/22/10 3:25:33 PM

A Beginner’s Guide to Structural Equation Modeling

5.1.1.2 Program
The next step is to create the LISREL–SIMPLIS program syntax file that will
specify the model analysis for Figure 5.1a. This is accomplished by selecting File on the tool bar, then clicking on New, select Syntax Only, and
then enter the program syntax. If you forget the SIMPLIS program syntax,
refer to the LISREL–SIMPLIS manual or modify an existing program. We
created a LISREL–SIMPLIS program named holz.spl that contains the following program syntax. (Note: The first three observed variables listed—
gender, ageyear, and birthmon—are contained in the raw data, but are not
analyzed in the SEM model.)
LISREL Figure 5.1a Program
Observed Variables
gender ageyear birthmon visperc cubes lozenges parcomp
sencomp wordmean
Correlation matrix from file holz.cor
Means from file holz.me
Standard deviations from file holz.sd
Sample Size 301

Y102005.indb 80

3/22/10 3:25:34 PM

Model Fit

Latent Variables
Spatial Verbal
Relationships
visperc - lozenges = Spatial
parcomp - wordmean = Verbal
Number of decimals = 5
Path Diagram
End of Problem

Select File, then Save As, to save the file as holz.spl (SIMPLIS file type).

You are now ready to run the analysis using the holz.spl file you just created. Click on the running L on the tool bar menu and the ASCII text file
holz.out will appear. The LISREL–SIMPLIS output file will contain several
model-fit indices; however, a LISREL–SIMPLIS program (holz.spl) and a
LISREL8 command program (holz.ls8) will report and use different chisquare fit values in the model-fit indices—that is, the minimum fit function chi-square (C1), the normal theory weighted least-squares fit function
(C2), the Satorra–Bentler scaled chi-square (C3), and the Browne adjusted
chi-square (C4) (see chapter note in Power and Sample Size section for
more detail).
5.1.1.3 Output
5.1.1.3.1 Goodness-of-Fit Statistics—Original Model
Degrees of Freedom = 8
Minimum Fit Function Chi-Square = 24.28099 (P = 0.0020559)
Normal Theory Weighted Least Squares Chi-Square = 24.40679
(P = 0.0019581)
Estimated Noncentrality Parameter (NCP) = 16.40679
90 Percent Confidence Interval for NCP = (5.18319 ; 35.23399)

Y102005.indb 81

3/22/10 3:25:34 PM

A Beginner’s Guide to Structural Equation Modeling

Minimum Fit Function Value = 0.080937
Population Discrepancy Function Value (F0) = 0.054689
90 Percent Confidence Interval for F0 = (0.017277 ; 0.11745)
Root Mean Square Error of Approximation (RMSEA) = 0.082681
90 Percent Confidence Interval for RMSEA = (0.046472 ; 0.12116)
P-Value for Test of Close Fit (RMSEA < 0.05) = 0.066396
Expected Cross-Validation Index (ECVI) = 0.16802
90 Percent Confidence Interval for ECVI = (0.13061 ; 0.23078)
ECVI for Saturated Model = 0.14000
ECVI for Independence Model = 2.49266
Chi-Square for Independence Model with 15 Degrees of Freedom =
735.79891
Independence AIC = 747.79891
Model AIC = 50.40679
Saturated AIC = 42.00000
Independence CAIC = 776.04157
Model CAIC = 111.59922
Saturated CAIC = 140.84932
Normed Fit Index (NFI) = 0.96700
Nonnormed Fit Index (NNFI) = 0.95765
Parsimony Normed Fit Index (PNFI) = 0.51573
Comparative Fit Index (CFI) = 0.97741
Incremental Fit Index (IFI) = 0.97763
Relative Fit Index (RFI) = 0.93813
Critical N (CN) = 249.24177
Root Mean Square Residual (RMR) = 2.01027
Standardized RMR = 0.047008
Goodness-of-Fit Index (GFI) = 0.97360
Adjusted Goodness-of-Fit Index (AGFI) = 0.93069
Parsimony Goodness-of-Fit Index (PGFI) = 0.37089
The chi-square statistic is significant, indicating a less-than-adequate model
fit to the sample variance–covariance matrix (Minimum Fit Function ChiSquare = 24.28099, df = 8, p = 0.0020559). Several of the other model-fit indices for the theoretical model in Figure 5.1a indicated a reasonable data
to model fit, for example, GFI = .97360, RMSEA = 0.082681, Standardized
RMR = .047008, and NFI = 0.96700. Modification indices in the computer
output, however, offer suggestions on how to further improve the model
to data-fit:

Y102005.indb 82

3/22/10 3:25:34 PM

Model Fit

The Modification Indices Suggest to Add the
Path to

from

visperc
lozenges
sencomp

Verbal
Verbal
Spatial

Decrease in Chi-Square

New Estimate

10.4
9.2
7.9

2.62
-2.32
-0.79

The Modification Indices Suggest to Add an Error Covariance
Between

and

cubes
lozenges
wordmean

visperc
cubes
parcomp

Decrease in Chi-Square

New Estimate

9.2
10.4
7.9

-8.53
8.59
-5.86

We wanted our theoretical model to keep Verbal and Spatial as separate
constructs (latent variables) with three separate sets of observed variables.
Therefore, we were not interested in adding any paths to either latent variable from the other latent variables observed variables. So, we choose to select
the adding of an error covariance between lozenges and cubes that would
decrease the model-fit chi-square value by an estimated 10.4. We, therefore,
added the following command line to our LISREL–SIMPLIS program:
Let the error covariance of lozenges and cubes correlate

Our modified theoretical model is diagrammed in Figure 5.1b. The
resulting computer output indicated a better model fit to the data with a
nonsignificant Minimum Fit Function c2 = 13.92604, df = 7, and p =.052513;
RMSEA = .056209; Standardized RMR = 0.032547, and GFI = .98508 . (Note:
We used a strict interpretation of p = .05 for model fit, so p = .053 was considered nonsignificant for model fit).
5.1.1.3.2 Goodness-of-Fit Statistics—Modified Model
Degrees of Freedom = 7
Minimum Fit Function Chi-Square = 13.92604 (P = 0.052513)
Normal Theory Weighted Least Squares Chi-Square = 13.63496 (P =
0.058068)
Estimated Noncentrality Parameter (NCP) = 6.63496
90 Percent Confidence Interval for NCP = (0.0 ; 21.19420)
Minimum Fit Function Value = 0.046420
Population Discrepancy Function Value (F0) = 0.022117
90 Percent Confidence Interval for F0 = (0.0 ; 0.070647)
Root Mean Square Error of Approximation (RMSEA) = 0.056209
90 Percent Confidence Interval for RMSEA = (0.0 ; 0.10046)
P-Value for Test of Close Fit (RMSEA < 0.05) = 0.35494

Y102005.indb 83

3/22/10 3:25:35 PM

A Beginner’s Guide to Structural Equation Modeling

0.09

visperc
0.96

0.90

cubes

0.20

0.31

Spatial

0.46
0.79

lozenges
0.42

0.27

parcomp

0.85

Verbal

0.86
0.27

sencomp

0.30

wordmean

0.84

Figure 5.1b
Modified common factor model. (From Holzinger, K. J., & Swineford, F. A. [1939]. A study
in factor analysis: The stability of a bi-factor solution. Supplementary Educational Monographs,
No. 48. Chicago: University of Chicago, Dept. of Education.)

Expected Cross-Validation Index (ECVI) = 0.13878
90 Percent Confidence Interval for ECVI = (0.11667 ; 0.18731)
ECVI for Saturated Model = 0.14000
ECVI for Independence Model = 2.49266
Chi-Square for Independence Model with 15 Degrees of Freedom =
735.79891
Independence AIC = 747.79891
Model AIC = 41.63496
Saturated AIC = 42.00000
Independence CAIC = 776.04157
Model CAIC = 107.53450
Saturated CAIC = 140.84932
Normed Fit Index (NFI) = 0.98107
Nonnormed Fit Index (NNFI) = 0.97941
Parsimony Normed Fit Index (PNFI) = 0.45783

Y102005.indb 84

3/22/10 3:25:35 PM

Model Fit

Comparative Fit Index (CFI) = 0.99039
Incremental Fit Index (IFI) = 0.99050
Relative Fit Index (RFI) = 0.95944
Critical N (CN) = 399.01152
Root Mean Square Residual (RMR) = 1.34928
Standardized RMR = 0.032547
Goodness-of-Fit Index (GFI) = 0.98508
Adjusted Goodness-of-Fit Index (AGFI) = 0.95523
Parsimony Goodness-of-Fit Index (PGFI) = 0.32836
Our LISREL–SIMPLIS example will further serve to help our understanding of how the various model-fit indices are computed and illustrate how
power and sample size can be determined. Overall, the fit indices fall into
the three main categories of model fit, model comparison, and model parsimony fit indices. Next, we discuss the fit indices in these three categories
to understand their development and recommended applications. [Note:
Extensive comparisons and discussions of many of these fit indices can
be found in issues of the following journals: Structural Equation Modeling:
A Multidisciplinary Journal, Psychological Bulletin, Psychological Methods, and
Multivariate Behavioral Research.]

5.2 Model Fit
Model fit determines the degree to which the sample variance–covariance
data fit the structural equation model. Model-fit criteria commonly used
are chi-square (c2), the goodness-of-fit index (GFI), the adjusted goodness-of-fit index (AGFI), and the root-mean-square residual index (RMR)
(Jöreskog & Sörbom, 1989). These criteria are based on differences between
the observed (original, S) and model-implied (reproduced, Σ) variance–
covariance matrices.
5.2.1 Chi-Square (c2)
A significant c2 value relative to the degrees of freedom indicates that
the observed and implied variance–covariance matrices differ. Statistical
significance indicates the probability that this difference is due to sampling variation. A nonsignificant c2 value indicates that the two matrices are similar, indicating that the implied theoretical model significantly
reproduces the sample variance–covariance relationships in the matrix.

Y102005.indb 85

3/22/10 3:25:35 PM

A Beginner’s Guide to Structural Equation Modeling

The researcher is interested in obtaining a nonsignificant c2 value with
associated degrees of freedom. Thus it may be more appropriate to call the
chi-square test a measure of badness-of-fit.
The chi-square test of model fit can lead to erroneous conclusions regarding analysis outcomes. The c2 model-fit criterion is sensitive to sample
size because as sample size increases (generally above 200), the c2 statistic
has a tendency to indicate a significant probability level. In contrast, as
sample size decreases (generally below 100), the c2 statistic indicates nonsignificant probability levels. The chi-square statistic is therefore affected
by sample size, as noted by its calculation, c2 = (n − 1) FML, where F is the
maximum likelihood (ML) fit function. The c2 statistic is also sensitive to
departures from multivariate normality of the observed variables.
Three estimation methods are commonly used to calculate c2 in latent
variable models (Loehlin, 1987): maximum likelihood (ML), generalized
least squares (GLS), and unweighted least squares (ULS). Each approach
estimates a best-fitting solution and evaluates the model fit. The ML estimates are consistent, unbiased, efficient, scale invariant, scale free, and
normally distributed if the observed variables meet the multivariate normality assumption. The GLS estimates have the same properties as the
ML approach under a less stringent multivariate normality assumption
and provide an approximate chi-square test of model fit to the data. The
ULS estimates do not depend on a normality distribution assumption;
however, the estimates are not as efficient, nor are they scale invariant or
scale free. The ML c2 statistic is c2 = (n − 1) FML, the GLS c2 statistic is c2 =
(n − 1) FGLS, and the ULS c2 statistic is c2 = (n − 1) FULS. (Note: see Chapter
Footnote.)
In our model analysis, we chose the maximum likelihood chi-square
estimation method (default setting). The ML c2 statistic uses the minimum fit function value, which is reported in the computer output. The
minimum fit function chi-square for our modified model is calculated as:
c2 = (301 − 1) .046420 = 13.926. (Note: add command line Number of decimals = 5
to SIMPLIS program so Minimum Fit Function Value = 0.046420 will not
differ due to rounding error.)
5.2.2 Goodness-of-Fit Index (GFI) and Adjusted
Goodness-of-Fit Index (AGFI)
The goodness-of-fit index (GFI) is based on the ratio of the sum of the
squared differences between the observed and reproduced matrices to
the observed variances, thus allowing for scale. The GFI measures the
amount of variance and covariance in S that is predicted by the reproduced matrix Σ. In our original model, GFI = .97, so 97% of the S matrix is
predicted by the reproduced matrix Σ, which improved in the modified
model to 99% where GFI = 0.98508.

Y102005.indb 86

3/22/10 3:25:35 PM

Model Fit

The GFI index can be computed for ML, GLS, or ULS estimates (Bollen,
1989). For our modified model the formula expression is:
GFI = 1 – [c2model/c2null]
GFI = 1 – [13.92604/735.79891]
GFI = 1 − .0189264
GFI = .98 ~ .99
( N ote : The c2null is the Chi-Square for Independence Model with 15
Degrees of Freedom.)

The adjusted goodness-of-fit index (AGFI) is adjusted for the degrees of
freedom of a model relative to the number of variables. The AGFI index is
computed as 1 − [(k/df) (1 − GFI)], where k is the number of unique distinct
values in S, which is p(p + 1)/2, and df is the number of degrees of freedom
in the model. The GFI index in our modified model analysis was .985,
therefore the AGFI index is
1 − [(k/df)(1 − GFI)] = 1 – [(15/7)(1 − .985)]
= 1 – [2.14285(.015)]
= 1 − .03
= .97
The GFI and AGFI indices can be used to compare the fit of two different
models with the same data or compare the fit of a single model using different data, such as separate datasets for males and females, for example,
or examine measurement invariance in group models.
5.2.3 Root-Mean-Square Residual Index (RMR)
The RMR index uses the square root of the mean-squared differences
between matrix elements in S and Σ. Because it has no defined acceptable
level, it is best used to compare the fit of two different models with the
same data. The RMR index is computed as
RMR = [(1/k) Σij (sij − σij)2]1/2.
For our example, the original model Root Mean Square Residual
(RMR) = 2.01027 compared to the modified model Root Mean Square
Residual (RMR) = 1.34928. There is also a standardized RMR, known as
Standardized RMR, which has an acceptable level when less than .05.
For our original model, the Standardized RMR = 0.047008, compared
to the modified model with a Standardized RMR = 0.032547, which is
deemed a more acceptable fit. (Note: The residual covariance matrix can

Y102005.indb 87

3/22/10 3:25:36 PM

A Beginner’s Guide to Structural Equation Modeling

be requested in a LISREL–SIMPLIS program by adding the command
line, Print Residuals.)

5.3 Model Comparison
Given the role chi-square has in the model fit of latent variable models,
three other indices have emerged as variants for comparing alternative
models: the Tucker–Lewis index (TLI) or Bentler–Bonett nonnormed
fit index (NNFI), the Bentler–Bonett normed fit index (NFI) (Bentler &
Bonett, 1980; Loehlin, 1987), and the comparative fit index (CFI). These
criteria typically compare a proposed model with a null model (independence model). In LISREL the null model is indicated by the independence-model chi-square value. The null model could also be any model
that establishes a baseline from which one could expect other alternative
models to be different.
5.3.1 Tucker–Lewis Index (TLI)
Tucker and Lewis (1973) initially developed the TLI for factor analysis
but later extended it to structural equation modeling. The measure can
be used to compare alternative models or to compare a proposed model
against a null model. The TLI is computed using the c2 statistic as
[(c2null /dfnull) − (c2proposed /dfproposed)]/[(c2null /dfnull) − 1]
It is scaled from 0 (no fit) to 1 (perfect fit). For our modified model analysis,
the NNFI, as it is known in LISREL, was computed as
Nonnormed Fit Index (NNFI)
= [(c2null /dfnull) – (c2proposed /dfproposed)]/[(c2null /dfnull) − 1]
= [(735.79891/15) – (13.92604 /7)]/[(735.7989/15) − 1]
= [(49.05326 – 1.98943)/(49.05326 – 1)]
= [47.06383/48.05326]
= 0.97941

5.3.2 Normed Fit Index (NFI) and Comparative Fit Index (CFI)
The NFI is a measure that rescales chi-square into a 0 (no fit) to 1.0 (perfect
fit) range (Bentler & Bonett, 1980). It is used to compare a restricted model

Y102005.indb 88

3/22/10 3:25:36 PM

Model Fit

with a full model using a baseline null model as follows: (c2null − c2model)/
c2null. In our modified model analysis this was computed as
Normed Fit Index (NFI) = (c2null − c2model)/c2null
= (735.7989 – 13.92604)/735.7989
= .98107
Bentler (1990) subsequently developed a coefficient of comparative fit
within the context of specifying a population parameter and distribution,
such as a population comparative fit index, to overcome the deficiencies
in NFI for nested models. The rationale for assessment of comparative fit
in the nested-model approach involves a series of models that range from
least restrictive (Mi) to saturated (Ms). Corresponding to this sequence of
nested models is a sequence of model-fit statistics with associated degrees
of freedom. The comparative fit index (CFI) measures the improvement in
noncentrality in going from model Mi to Mk (the theoretical model) and
uses the noncentral c2 (dk) distribution with noncentrality parameter l k
to define comparative fit as (l i − l k)/l i. In our modified model output the
Comparative Fit Index (CFI) = 0.99039.
McDonald and Marsh (1990) further explored the noncentrality and
model-fit issue by examining nine fit indices as functions of noncentrality
and sample size. They concluded that only the Tucker-Lewis Index and
their relative noncentrality index (RNI) were unbiased in finite samples
and recommended them for testing null or alternative models. For absolute measures of fit that do not test null or alternative models, they recommended dk (Steiger & Lind, 1980), because it is a linear function of c2, or
a normed measure of centrality mk (McDonald, 1989), because neither of
these varies systematically with sample size. These model fit measures
of centrality are useful when selecting among a few competing models
based upon theoretical considerations.

5.4 Model Parsimony
Parsimony refers to the number of estimated parameters required to
achieve a specific level of model fit. Basically, an over-identified model
is compared with a restricted model. The AGFI measure discussed previously also provides an index of model parsimony. Other indices that
indicate model parsimony are the parsimony normed fit index (PNFI),
and the Akaike information criterion (AIC). Parsimony-based fit indices

Y102005.indb 89

3/22/10 3:25:36 PM

A Beginner’s Guide to Structural Equation Modeling

for multiple indicator models were reviewed by Williams and Holahan
(1994). They found that the AIC performed the best (see their article for
more details on additional indices and related references). The model parsimony goodness-of-fit indices take into account the number of parameters required to achieve a given value for chi-square. Lower values for
PNFI and AIC indicate a better model fit given a specified number of
parameters in a model.
5.4.1 Parsimony Normed Fit Index (PNFI)
The PNFI measure is a modification of the NFI measure (James, Mulaik, &
Brett, 1982). The PNFI, however, takes into account the number of degrees
of freedom used to obtain a given level of fit. Parsimony is achieved with
a high degree of fit for fewer degrees of freedom in specifying the coefficients to be estimated. The PNFI is used to compare models with different
degrees of freedom and is calculated as PNFI = (dfproposed /dfnull) NFI. In our
modified model analysis the PNFI was:
Parsimony Normed Fit Index (PNFI) = (dfproposed/dfnull) NFI
= (7/15) .98107
= 0.45783
5.4.2 Akaike Information Criterion (AIC)
The AIC measure is used to compare models with differing numbers of
latent variables, much as the PNFI is used (Akaike, 1987). The AIC can be
calculated in two different ways: c2 + 2q, where q = number of free parameters in the model, or as c2 – 2df. The first AIC is positive (as computed
in LISREL), and the second AIC is negative, but either AIC value close to
zero indicates a more parsimonious model. The AIC indicates model fit (S
and Σ elements similar) and model parsimony (over-identified model). In
our modified model analysis, the computer output gives several AIC values for the theoretical model, saturated model, and independence model;
however, we only report two AIC fit indices. (Note: AIC uses Normal
theory weighted least squares chi-square not the minimum fit function
chi-square.)
Model AIC = Normal Theory c2 + 2q
= 13.63496 + 2 (14)
= 41.63496

Y102005.indb 90

3/22/10 3:25:36 PM

Model Fit

Independence AIC = Chi-Square for Independence Model + 2(df -1)
= 735.79891 + 2(6)
= 747.79891

5.4.3 Summary
Mulaik, James, Alstine, Bennett, Lind, and Stilwell (1989) evaluated the
c2, NFI, GFI, AGFI, and AIC goodness-of-fit indices. They concluded that
these indices fail to assess parsimony and are insensitive to misspecification of structural relationships (see their definitive work for additional
information). Their findings should not be surprising because it has been
suggested that a good fit index is one that is independent of sample size,
accurately reflects differences in fit, imposes a penalty for inclusion of
additional parameters (Marsh, Balla, & McDonald, 1988), and supports the
choice of the true model when it is known (McDonald & Marsh, 1990). No
model-fit criteria can actually meet all of these criteria.
We have presented several model-fit indices that are used to assess
model fit, model comparison, or model parsimony. In addition, we calculated many of these based on the model analyzed in this chapter. The
LISREL program outputs many different model-fit criteria because more
than one should be reported. The LISREL user guides also provide an
excellent discussion of the model-fit indices in their program. We recommend that once you feel comfortable using these fit indices for your
specific model applications, you check the references cited for additional
information on their usefulness and/or limitations. Following their initial description, there has been much controversy and discussion on
their subjective interpretation and appropriateness under specific modeling conditions (see Marsh, Balla, & Hau [1996] for further discussion).
Further research and discussion will surely follow; for example, Kenny
& McCoach (2003) indicated that RMSEA improves as more variables are
added to a model, whereas TLI and CFI both decline in correctly specified
models as more variables are added.
When deciding on which model-fit indices to report, first consider
whether the fit indices were created for model fit, model parsimony, or
model comparison. At the risk of oversimplification, we suggest that c2,
RMSEA, and Standardized RMR be reported for all types of models with
additional fit indices reported based on purpose of modeling. For example, the CFI should be reported if comparing models. Overall, more than
one model-fit index should be reported. If a majority of the fit indices on
your list indicate an acceptable model, then your theoretical model is supported by the data.

Y102005.indb 91

3/22/10 3:25:36 PM

A Beginner’s Guide to Structural Equation Modeling

5.5 Parameter Fit
Individual parameter estimates in a model can be meaningless even though
model-fit criteria indicate an acceptable measurement or structural model.
Therefore, interpretation of parameter estimates in any model analysis is
essential. The following steps are therefore recommended:
1. Examine the parameter estimates to determine whether they have
the correct sign (either positive or negative).
2. Examine parameter estimates (standardized coefficients) to determine whether they are out of bounds or exceed an expected range
of values.
3. Examine the parameter estimates for statistical significance
(T or Z-values = parameter estimate divided by standard error of
parameter estimate).
4. Test for measurement invariance by setting parameter estimates
equal (constraints) in different groups, for example, girls and
boys, then make relative comparisons among the parameter
estimates.
An examination of initial parameter estimates can also help in identifying a faulty or misspecified model. In this instance, initial parameter estimates can serve as start values—for example, initial two-stage
least-squares (TSLS) estimates in LISREL. The researcher then replaces
the TSLS estimate with a user-defined start value. Sometimes parameter estimates take on impossible values, as in the case of a correlation
between two variables that exceeds 1.0. Sometimes negative variance is
encountered (known as a Heywood case). Also, if the error variance for
a variable is near zero, the indicator variable implies an almost perfect
measure of the latent variable, which may not be the case. Outliers can
also influence parameter estimates. Use of sufficient sample size (n > 100
or 150) and several indicators per latent variable (four is recommended
based on the TETRAD approach) has also been recommended to produce reasonable and stable parameter estimates (Anderson & Gerbing,
1984).
Once these issues have been taken into consideration, the interpretation of modification indices and expected parameter change can begin
to modify the model, but there is still a need for guidance provided by
the rationale for the theoretical model and the researcher’s expertise.
Researchers should use the model-fit indices as potential indicators of
misfit when respecifying or modifying a model. Cross-validation or

Y102005.indb 92

3/22/10 3:25:37 PM

Model Fit

replication using another independent sample, once an acceptable model
is achieved, is always recommended to ensure stability of parameter
estimates and validity of the model (Cliff, 1983). Bootstrap procedures
also afford a resampling method, given a single sample, to determine
the efficiency and precision of sample estimates (Lunneborg, 1987).
These model validation topics are discussed further in chapter 12.

5.6 Power and Sample Size
The determination of power and/or sample size in SEM is complicated
because theoretical models can have several variables or parameter estimates and parameters are typically not independent in a model and have
different standard errors. In SEM we also compare models, oftentimes
nested models with the same data set. Consequently, power and sample
size determination in the situation where a researcher is hypothesis testing (testing a model fit to data), comparing alternative models, or desiring
to test a parameter estimate for significance will be covered with SAS,
SPSS, and G*Power 3 examples using the LISREL–SIMPLIS example in the
chapter. The power for hypothesis testing, or the probability of rejecting Ho
when Ha is true, depends on the true population model, significance level,
degrees of freedom, and sample size, which involves specifying an effect
size, alpha, and sample size; while sample size determination is achieved
given power, effect size, and alpha level of significance. Daniel S. Soper
has a user friendly website that provides effect size, power and sample
size determination in statistics (http://www.danielsoper.com/statcalc/).
Hypothesis testing involves confirming that a theoretical model fits the
sample variance–covariance data, comparing fit between alternative models, or testing parameter coefficients for significance; even whether coefficients are equal between groups. These hypothesis testing methods should
involve constrained models with fewer parameters than the initial model.
The initial (full) model represents the null hypothesis (Ho) and the alternative (constrained) model with fewer parameters is denoted Ha. Each model
generates a c2 goodness-of-fit measure, and the difference between the
models for significance testing is computed as D2 = c2o − c2a, with dfd =
dfo − dfa. The D2 statistic is tested for significance at a specified alpha level
(probability of Type I error), where Ho is rejected if D2 exceeds the critical
tabled c2 value with dfd degrees of freedom (Table A.4). The chi-square difference test or likelihood ratio test is used with GLS, ML, and WLS estimation methods.

Y102005.indb 93

3/22/10 3:25:37 PM

A Beginner’s Guide to Structural Equation Modeling

The significance of parameter estimates that do not require two separate models to yield separate c2 values includes: (a) generating a twosided t or z value for the parameter estimate (T or Z = parameter estimate
divided by standard error of the parameter estimate), and (b) interpreting the modification index directly for the parameter estimate as a c2
test with 1 degree of freedom. The relationship is simply T2 = D 2 = MI
(modification index) for large sample sizes. Gonzalez and Griffin (2001),
however, indicated that the standard errors of the parameter estimates
are sensitive to how the model is identified, that is, alternative ways of
identifying a model may yield different standard errors, and hence different T values for the statistical significance of a parameter estimate.
This lack of invariance due to model identification could result in different conclusions about a parameter’s significance level from different,
yet equivalent, models on the same data. The authors recommended
that parameter estimates be tested for significance using the likelihood
ratio (LR) test because it is invariant to model identification, rather than
the T test (or z test).
5.6.1 Model Fit
A traditional approach in SEM is to hypothesize a theoretical model, collect sample data, and test whether the model fits the data. In this chapter we have discussed various fit indices to determine if the theoretical
model fits the data. When the theoretical model does not fit the data, we
look to modification indices for suggestions on how to modify the model
for an improved fit. The power to reject a null hypothesis and sample
size impacts our decision of whether sample data fit a theoretical model.
Power and sample size are therefore discussed next.
5.6.1.1 Power
Saris and Satorra (1993) provided an easy to use approach for calculating
power of a theoretical model. Basically, an alternative model is estimated
with sample data to indicate what percent of the time we would correctly
reject the null hypothesis under the assumption that the null hypothesis
(Ho) is false. The minimum fit function chi-square value obtained from
fitting data to the theoretical model provides an estimate of the noncentrality parameter (NCP). NCP is calculated as Normal Theory Weighted
Least Squares c2 – dfmodel. For our modified model the NCP = 13.63496 – 7
= 6.63496, which is provided in the Goodness-of-Fit section of the computer output. This makes computing power using SAS 9.1, SPSS 16.0, or
G*Power 3 straightforward, using their respective command functions.

Y102005.indb 94

3/22/10 3:25:37 PM

Model Fit

(Note: c2 = 3.841, df = 1, p = .05 is the critical tabled value for testing
our hypothesis of model fit.) Examples for each using NCP are provided
next.
SAS syntax—power
data chapter5;
do obs=1;
ncp = 6.63496;
power = 1 – PROBCHI(3.841, 1, ncp);
output;
end;
proc print;
var ncp power;
run;

SPSS syntax—power
DATA LIST FREE / obs.
BEGIN DATA.
1
END DATA.
compute ncp = 6.63496.
compute power = 1 - NCDF.CHISQ(3.841, 1, ncp).
formats ncp power (f8.5).
List.

In our modified model, NCP = 6.63496, so our power = .73; the output from
the SAS or SPSS syntax was:
obs
1.00

ncp
6.63496

power
.73105

Power, given your model fit, can also be determined using G*power 3 (Faul,
Erdfelder, Lang & Buchner, 2007). The free G*Power 3 software download
is available from the Web site: http://www.psycho.uni-duesseldorf.de/
abteilungen/aap/gpower3/, which is somewhat easier than running the
SAS and SPSS programs. Power and sample size estimates for a priori and
post-hoc statistical applications are available using G*power 3. (Note: We
used G*Power 3, Windows, Release 3.1.0, 2008, but a MAC OS version is
also available). After download and installation, click on the G*Power 3
desktop icon and you should see the following dialog box:

Y102005.indb 95

3/22/10 3:25:37 PM

A Beginner’s Guide to Structural Equation Modeling

In the Test family window select “c2 tests”; in the Statistical test window
select “Generic c2 test”; and in the Type of power analysis window, select
“Post-hoc: Compute power – given a, and noncentrality parameter.” Our modified model had NCP = 6.63496, so we entered this value in the “noncentrality parameter l” window along with df = 1 and a = .05. The dialog box
should look like:

Y102005.indb 96

3/22/10 3:25:38 PM

Model Fit

Click the Calculate button. The power = .731015 value matches our earlier
calculations. The dialog box should now look like:

Y102005.indb 97

3/22/10 3:25:39 PM

A Beginner’s Guide to Structural Equation Modeling

Our modified model has a 73 percent chance of rejecting the null
hypothesis at the .05 level of significance, which falls short of the .80 level
commonly accepted for power. If we replace the critical chi-square value
in the formula, you can determine power for other alpha levels of significance. In Table 5.2, we have replaced the critical chi-square value and ran
the SPSS syntax program for alpha values ranging from .10 to .001. If we
test our modified model fit at the p = .10 level, then we achieve an acceptable level of power; other alpha levels from .05 to .001 fall below a .80
power value.

Y102005.indb 98

3/22/10 3:25:40 PM

Model Fit

Table 5.2
Power for Alpha Levels Given Modified Model NCP
NCP
6.63496
6.63496
6.63496
6.63496
6.63496

Critical Chi-Square

Alpha

Power

2.706
3.841
5.412
6.635
10.827

.10
.05
.02
.01
.001

.82405
.73105
.59850
.50000
.23743

Note: Critical c2 values for df = 1 from Table A.4.

5.6.1.2 Sample Size
An earlier way to determine an appropriate sample size in SEM was given
by Hoelter (1983) as the Critical N (CN) statistic, where CN ≥ 200 was considered adequate. The Critical N is calculated as:
CN = (c2critical/Fmin) + 1
The critical chi-square (c2critical) is obtained for the model degrees of
freedom at the .05 level of significance. The CN statistic is output by the
LISREL–SIMPLIS program. In our final modified model of Figure 5.1b,
CN = 399, Fmin was determined to be .0353432 and c2critical = 14.067 for df =
7 at .05 level of significance (see Table A.4); so CN = (14.067/.0353432) + 1 =
399. (Note: our modified model computer output indicated Fmin = .04642,
p = .0525, but CN uses Fmin at p = .05). CN gives the sample size at which
the Fmin value leads to a rejection of Ho. Our sample size was N = 301
with a nonsignificant chi-square (minimum fit function c2, p = .052; or normal theory weighted least squares c2, p = .058) and several good model-fit
indices, so even if we used Hoelter’s suggestion, we had sufficient sample
size. For a further discussion about CN refer to Bollen and Liang (1988) or
Bollen (1989).
Sample size influences the calculation of the minimum fit function c2.
Recall that the Minimum Fit Function c2 in the modified model was computed as:
Minimum Fit Function c2 = (N – g) × Fmin
= (301 – 1) × (.046420)
= 13.92604

Y102005.indb 99

3/22/10 3:25:40 PM

100

A Beginner’s Guide to Structural Equation Modeling

Sample size also influences the calculation of the Fmin values as follows:
Fmin = Minimum Fit Function c2/(N – g)
= 13.92604/(301 – 1)
= .046420
The Fmin is computed using the minimum fit function c2 in the computer
output, sample size (N), and number of groups (g); while the noncentrality parameter (NCP) is computed using the Normal Theory c2 minus the
degrees of freedom in the model. NCP is therefore computed as:
NCP = Normal Theory Weighted Least Squares c2 – dfmodel
= 13.63496 – 7
= 6.63496
Estimated sample size (N) using these NCP and Fmin values is less than
our actual sample size of N = 301:
N = (NCP/Fmin) + g
= (6.63496/.046420) + 1
= 143.93 ~ 144
You have probably noticed that Fmin is calculated using the Minimum Fit
Function c2, but NCP is calculated using the Normal Theory Weighted
Least Squares c2. LISREL, unlike other SEM software calculates some
measures of fit (NCP, RMSEA, and Independence model c2 ) using the
normal theory weighted least squares c2, but uses the minimum fit
function c2 for others. Differences between these two can be small if
the multivariate normality assumption holds or very different if not (see
Chapter Footnote for detailed description of standard errors and four
different c2 values: C1 = minimum fit function c2; C2 = normal theory
weighted least squares c2; C3 = Satorra–Bentler scaled c2; C4 = c2 corrected for nonnormality).
To determine sample size for given df, alpha, and power for a theoretical
model, the Fmin value would be fixed (Fmin value from your final model; our
modified model had Fmin = .046420), but the NCP value would vary. For
our modified model, the SAS program can be run for differing NCP values to obtain corresponding sample size and power estimates. (Note: We

Y102005.indb 100

3/22/10 3:25:40 PM

101

Model Fit

are changing values of power in the SAS syntax program, but you can also
fix power and change alpha values to obtain different sample sizes for different alpha levels at a specified power level, for example, power = .80).
SAS syntax—sample size
data chapter5;
do obs = 1;
g = 1;
* change values of alpha to obtain sample size for given
power;
alpha = .05;
fmin = .046420;
df = 1;
* change values of power to obtain sample size for given
alpha;
power = .60;
chicrit = quantile(‘chisquare’,1 – alpha, df);
ncp = CINV(power,df,chicrit);
n = (ncp/fmin) + g;
output;
end;
proc print;
var power n alpha ncp fmin g;
run;

The output from this first run with power = .60 would look like this:
obs
1

power
n
alpha
ncp
fmin
.6
106.535
.05
4.89892 .04642

g
1

chicrit
3.84146

We created Table 5.3 by changing the value of power for alpha = .05
for a critical c2 = 3.841, df = 1. (Note: fmin is fixed at the value from our

Table 5.3
Sample Size for Given Power with Alpha = .05
Power
.60
.70
.73
.80
.90
.95

Alpha

ncp

fmin

c2 critical

106.535
133.963
143.594
170.084
227.356
280.938

.05
.05
.05
.05
.05
.05

4.89892
6.17213
6.61923
7.84890
10.5074
12.9947

.04642
.04642
.04642
.04642
.04642
.04642

1
1
1
1
1
1

3.84146
3.84146
3.84146
3.84146
3.84146
3.84146

Note: n should be rounded up, for example, 106.535 = 107.

Y102005.indb 101

3/22/10 3:25:40 PM

102

A Beginner’s Guide to Structural Equation Modeling

modified model; alpha is fixed at .05, so chicrit will be fixed at 3.84146).
A sample size of N = 144 for power = .73 from our modified model was
also correctly computed and indicated in the table. We see in Table 5.3
that sample size requirements increase as power increases, which is
expected.
In our modified model we have N = 301, NCP = 6.63496, and our posthoc power = .73 calculated at the .05 level of significance. A sample size of
N = 170 would have given us power = .80 at the .05 level of significance.
Are you puzzled? Well, recall that NCP = c2 – dfmodel, so if our model had
resulted in a NCP = 7.84890 with N = 170 at the .05 level of significance,
then we would have achieved an acceptable level of power = .80. We find
that the noncentrality parameter (NCP) is affected by the model chisquare but also the degrees of freedom, which indicates a certain level of
model complexity.
We can also use the SAS syntax—sample size program to examine how
changing the level of significance affects sample size for a fixed power
value. Recall that Fmin is fixed at .04642 from our modified model. Table 5.4
contains the output from the SAS program. We see in Table 5.4 that sample
size requirements increase as the level of significance (alpha) for testing
our model decreases, which is expected.
Table 5.4
Sample Size for Given Alpha with Power = .80
Power
.8
.8
.8
.8
.8

Alpha

ncp

fmin

c2 critical

134.194
170.084
217.201
252.593
368.830

.10
.05
.02
.01
.001

6.18288
7.84890
10.0360
11.6790
17.0746

.04642
.04642
.04642
.04642
.04642

1
1
1
1
1

2.70554
3.84146
5.41189
6.63490
10.8276

Note: c2critical values correspond to alpha values in Table A.4.

We used G*Power 3 to calculate various NCP values given alpha and
power because SPSS 16.0 does not have a command function at this
time to determine the noncentrality parameter (NCP) given power,
df, and critical c2. (Note: SAS, S-Plus, Stata and other statistical software have this capability) In the Test family drop-down menu, select
“c2 test”; in the Statistical Test drop-down menu select, “Generic c2 test”;
and in the Type of power analysis, select “Sensitivity: Compute noncentrality parameter – given a, and power.” In the Input Parameters boxes, change

Y102005.indb 102

3/22/10 3:25:40 PM

Model Fit

103

the power value to .80 and the df value to 1. Your dialog box should
now appear as:

Click on the Calculate button; the Output Parameters, “Critical c2” and
“Noncentrality parameter l“ will appear. The G*Power 3.1.0 dialog box will
now display the Critical c2 = 3.84146 (associated with alpha = .05, df = 1)
and corresponding noncentrality parameter for power = .80. Your dialog
box should now look like:

Y102005.indb 103

3/22/10 3:25:41 PM

104

A Beginner’s Guide to Structural Equation Modeling

Table 5.3 reports these same values using the program SAS syntax—
sample size.
You can click on the X-Y plot for a range of values button to enter a range of
power values that can be plotted by corresponding noncentrality parameter values. (Note: Check the box for “and displaying the values in the plot”
and change the “in steps of” from .01 to .10 for clarity in the output of the
graph.) The dialog box should look like this:

Y102005.indb 104

3/22/10 3:25:41 PM

Model Fit

105

Click on Draw plot. Your graph will now appear and should look like the
dialog box below:

Y102005.indb 105

3/22/10 3:25:42 PM

106

A Beginner’s Guide to Structural Equation Modeling

A range of power values entered from .6 to .95 indicates the levels
of the noncentrality parameter that one needs to exceed for that level of
power [see Table 5.3].)

(Note :

In planning a study, we should determine a priori what our sample size
and power values should be. After gathering our data and running our SEM
model (and any modifications), we should compute the post-hoc power using
our noncentrality parameter from the LISREL computer output or sample
size (N) using NCP and model Fmin values. This should be easy given that
N = (NCP/Fmin) + g. We can a priori specify values or obtain the Fmin value
from our model, calculate NCP using SAS or G*power 3 for a given df, critical chi-square, power, then use these values to calculate sample size (N).
MacCallum, Browne, and Sugawara (1996) provided a different approach to
testing model-fit using the root mean square error of approximation (RMSEA).
Their approach also emphasized confidence intervals around RMSEA, rather
than a single point estimate, so they suggested null and alternative values for
RMSEA (exact fit: Ho = .00 versus Ha = .05; Close fit: Ho = .05 versus Ha = .08;
and Not close fit: Ho = .05 versus Ha = .10); researchers can also select their
own. The MacCallum et al. (1996) method tests power, given exact fit (Ho;
RMSEA = 0), close fit (Ho, RMSEA ≤ .05), or not close fit (Ho, RMSEA ≥ .05); and
included SAS programs for calculating power given sample size or sample
size given power using RMSEA. RMSEA is calculated as:
RMSEA = ( NCP/N − 1)/df
For our modified model, NCP = 6.63496; N = 301; and df = 7, so RMSEA =
0.056209:
RMSEA = (6.63496/300)/7 = .056209
SAS syntax—RMSEA and power
data chapter5;
do obs = 1;
n = 301;
df = 7;
alpha = .05;
* change rmseaHo and rmseaHa values to correspond to exact,
close, and not close values;
rmseaHo = .05;
rmseaHa = .08;
ncpHo = (n-1)*df*rmseaHo*rmseaHo;
ncpHa = (n-1)*df*rmseaHa*rmseaHa;
chicrit = quantile(‘chisquare’,1-alpha,df);
if rmseaHo < rmseaHa then power = 1 –
PROBCHI(chicrit,df,ncpHa);

Y102005.indb 106

3/22/10 3:25:43 PM

Model Fit

107

if rmseaHo > rmseaHa then power = PROBCHI(chicrit,df,ncpHa);
output;
end;
Proc print;
Var n df alpha rmseaHo rmseaHa ncpHo ncpHa chicrit power;
Run;

SPSS syntax—RMSEA and power
DATA LIST FREE / obs.
BEGIN DATA.
1
END DATA.
compute n = 301.
compute df = 7.
compute alpha = .05.
comment change rmseaHo and rmseaHa values to correspond with
exact, close, not close values.
compute rmseaHo = .05.
compute rmseaHa = .08.
compute ncpHo = (n-1)*df*rmseaHo*rmseaHo.
compute ncpHa = (n-1)*df*rmseaHa*rmseaHa.
compute chicrit = IDF.CHISQ(1-alpha,df).
do if (rmseaHo < rmseaHa).
compute power = 1 - NCDF.CHISQ(chicrit, df, ncpHa).
else if (rmseaHo > rmseaHa).
compute power = NCDF.CHISQ(chicrit, df, ncpHa).
end if.
formats chicrit ncpHo ncpHa power (f8.5).
List.

The resulting SAS or SPSS output for close fit was given as:
obs
1.00

n df alpha rmseaHo rmseaHa ncpHo ncpHa chicrit power
301 7 .05
.05
.08
5.25 13.44 14.0671 .76813

We ran the recommended RMSEA values given by MacCallum et al. (1996)
and listed them in Table 5.5. For exact fit, power = .33, for close fit, power =
.76, and for not close fit, power = .06 ~ .057. A RMSEA model-fit value
between .05 and .08 is considered an acceptable model-fit index, when
reported with other fit indices. Our modified model RMSEA = .056209
and for close fit had power = .76813.

Y102005.indb 107

3/22/10 3:25:43 PM

108

A Beginner’s Guide to Structural Equation Modeling

Table 5.5
MacCallum et al. (1996) Null and Alternative Values for
RMSEA Test of Fit
MacCallum Test

Power

Exact
Close
Not Close

.00
.05
.05

.05
.08
.01

.33034
.76813
.05756

Figure 5.1, Modified model (a = .05, df = 7, N = 301).

5.6.2 Model Comparison
A likelihood ratio (LR) test is possible between alternative models to
examine the difference in c2 values between the initial model and a modified model. The LR test with degrees of freedom equal to df Initial − df Modified
is calculated as:
LR = c2Initial − c2Modified
For our example, the initial model had c2 = 24.28099, df = 8, and the modified model had c2 = 13.92604, df = 7; therefore, LR = 10.35495 with df = 1,
which is a statistically significant chi-square value at the .05 level of significance (c2 > 3.84, df = 1, a = .05), indicating the models are different.
LRdf = 1 = 24.28099 – 13.92604 = 10.35495
The LR test between models is possible when adding or dropping a single
parameter (path or variable). In LISREL–SIMPLIS, a researcher will most
likely be guided by the modification indices with their associated change
(decrease) in chi-square when respecifying or modifying a model. On the
basis of our LISREL–SIMPLIS modification indices, we chose to add an
error covariance between lozenges and cubes by adding the following command in our subsequent model analysis because it gave us our largest
decrease in model chi-square (see Figure 5.1b):
Let the error covariance of lozenges and cubes correlate
MacCallum, Browne, and Cai (2006) presented an approach to compare
nested models when the between model degrees of freedom are ≥ 1. They
showed that when testing close fit, power results may differ depending
upon the degrees of freedom in each model. Basically, the power to detect
differences will be greater when models being compared have more
degrees of freedom. For any given sample size, power increases as the
model degrees of freedom increases. They defined an effect size (d) in
terms of model RMSEA and degrees of freedom for the two models, so in

Y102005.indb 108

3/22/10 3:25:43 PM

109

Model Fit

our example, the effect size (d) would be computed as:
d = (dfInitial * RMSEA2Initial – dfModified * RMSEA2Modified)
d = ([8 * (.080937)2] – [7 * (.046420)2])
d = (.0524056 − .0150836)
d = .037322
The noncentrality parameter is computed as:
NCP = (N – 1) d
So, for our example:
NCP = (301 – 1) * ( .037322)
NCP = 11.1966
Using G*Power 3, we enter this NCP = 11.1966, .05 level of significance, and
df = 1 (model degree of freedom difference) and obtained power = .917. The
G*Power 3 dialog box should look like this:

Y102005.indb 109

3/22/10 3:25:44 PM

110

A Beginner’s Guide to Structural Equation Modeling

Power to detect a difference in RMSEA values is therefore possible for
a given sample size with various degrees of freedom. The SAS program
below will also provide an ability to make power comparisons for different model degrees of freedom using RMSEA values from two nested
models.
SAS syntax—effect size, RMSEA, and power
data chapter5;
do obs = 1;
n = 301;
alpha = .05;
dfa = 8;
rmseaA = .080937;
dfb = 7;
rmseaB = .046420;
delta = (dfa*rmseaA*rmseaA) – (dfb*rmseaB*rmseaB);
ncp = (n – 1)*delta;
dfdiff = dfa – dfb;
chicrit = quantile(‘chisquare’,1 – alpha, dfdiff);
power = 1 – PROBCHI(chicrit, dfdiff,ncp);
output;
end;
Proc print;
var n dfa rmseaA dfb rmseaB delta ncp dfdiff chicrit power;
run;

The computer output should look like this:
Obs
1
delta
.037323

n
301
ncp
11.1968

dfa
8
dfdiff
1

rmseaA
0.080937
chicrit
3.84146

dfb
7

rmseaB
0.04642

power
.91716

The power = .91716 indicates a 91% chance of detecting a difference
between the model RMSEA values.
Power is affected by the size of the model degrees of freedom (degrees
of freedom implies a certain degree of model complexity). The G*Power 3
program or the SAS program can be used for models where the difference
in degrees of freedom is greater than one. We therefore ran a comparison
for our model with different levels of degrees of freedom to show how
power is affected. In Table 5.6, power increases dramatically when the
level of degrees of freedom increases from 5 to 14 while maintaining a
model degrees of freedom difference at df = 1. You can also output program values for df ≥ 2 to see effect on power.

Y102005.indb 110

3/22/10 3:25:44 PM

111

Model Fit

Table 5.6
MacCallum et al. (2006) Power
at Increasing Model Degrees of
Freedom
dfa

dfb

Power

5
8
11
14

4
7
10
13

.76756
.91716*
.97337
.99206

(RMSEA approach)
Figure 5.1b Model (a = .05, N = 301).

5.6.3 Parameter Significance
A single parameter can be tested for significance using nested models.
Nested models involve an initial model being compared to a modified model
in which a single parameter has been fixed to zero (dropped) or estimated
(added). In structural equation modeling, the intent is to determine the
significance of the decrease in the c2 value for the modified model from
the initial model. The LR test was used before to test the difference in the
models for our single parameter that we added (error covariance between
lozenges and cubes).
Power can be computed for testing the significance of an individual
parameter estimate. For GLS, ML, and WLS estimation methods, this
involves determining the significance of c2 with one degree of freedom
(c2 > 3.84, df = 1, a = .05) for a single parameter estimate, thus determining the significance of the reduction in c2 that should equal or exceed the
modification index value for the parameter estimate fixed to zero. Power
values for modification index values can be computed using SAS because
the modification index (MI) is a noncentrality parameter (NCP). The
power of a MI value (NCP) at the .05 level of significance, df = 1, critical
chi-square value = 3.841 is computed in the following SAS syntax program for our MI = 10.4 when adding the error covariance between lozenges
and cubes. Power = .89, so in testing the statistical significance of MI for our
parameter (error covariance), we have an 89% chance of correctly rejecting the null hypothesis and accepting the alternative hypothesis that MI
is different from zero.
SAS syntax—power for parameter MI value
data chapter5;
do obs = 1;
mi = 10.4;
alpha = .05;

Y102005.indb 111

3/22/10 3:25:44 PM

112

A Beginner’s Guide to Structural Equation Modeling

df = 1;
chicrit = quantile(‘chisquare’,1 – alpha, df);
power = 1 – PROBCHI(chicrit, df, mi);
output;
end;
Proc print;
var mi power alpha df chicrit;
run;

The SAS output indicated the following:
Obs
1

mi
10.4

power
.89075

alpha
.05

df
1

chicrit
3.84146

Power values for parameter estimates can also be computed using a
SAS program because a squared T or Z value for a parameter estimate is
asymptotically distributed as a noncentral chi-square, that is, NCP = T2.
Our modified model indicated an error covariance = 8.34 (modification
index indicated a New Estimate at 8.59), with standard error = 2.62, so
T = 8.34/2.62 = 3.19; LISREL program output provided these values for
the added parameter:
Error Covariance for lozenges and cubes = 8.34
(2.62)
3.19
(Note : LISREL 8.8 student version lists standared error, Z value, and
p-value in the output)

The power of a squared T value for our parameter estimate is computed in
a SAS program as follows:
SAS syntax—power for parameter T value
data chapter5;
do obs = 1;
T = 3.19;
ncp = T*T;
alpha = .05;
df = 1;
chicrit = quantile(‘chisquare’,1 – alpha, df);
power = 1 – PROBCHI(chicrit, df, ncp);
output;
end;
Proc print;
var ncp power alpha df chicrit;
run;

Y102005.indb 112

3/22/10 3:25:44 PM

113

Model Fit

The SAS output looks like this:
Obs
1

ncp
10.1761

power
.89066

alpha
.05

df
1

chicrit
3.84146

Power = .89, so in testing the statistical significance of our parameter estimate, we have an 89% chance of correctly rejecting the null hypothesis and
accepting the alternative hypothesis that T is different from zero. (Note: The
other model-fit indices [GFI, AGFI, NFI, IFI, CFI, etc.] do not have a test of
statistical significance and therefore do not involve power calculations).
5.6.4 Summary
Research suggests that certain model-fit indices are more susceptible to
sample size than others, hence, power. We have already learned that c2 is
affected by sample size, that is, c2 = (N − 1) FML , where FML is the maximum
likelihood fit function for a model, and therefore c2 increases in direct relation to N − 1 (Bollen, 1989). Kaplan (1995) also pointed out that power in
SEM is affected by the size of the misspecified parameter, sample size, and
location of the parameter in the model. Specification errors induce bias
in the standard errors and parameter estimates, and thus affect power.
These factors also affect power in other parametric statistical tests (Cohen,
1988). Saris and Satorra (1993) pointed out that the larger the noncentrality
parameter, the greater is the power of the test, that is, an evaluation of the
power of the test is an evaluation of the noncentrality parameter.
Muthén and Muthén (2002) outlined how Monte Carlo methods can be
used to decide on the power for a given specified model using the Mplus
program. Power is indicated as the percentage of significant coefficients or
the proportion of replications for which the null hypothesis that a parameter is equal to zero is rejected at the .05 level of significance, two-tailed
test, with a critical value of 1.96. The authors suggested that power equal
or exceed the traditional .80 level for determining the probability of rejecting the null hypothesis when it is false.
Marsh et al. (1988, 1996) also examined the influence of sample size on 30
different model-fit indices and found that the Tucker–Lewis index (Tucker
& Lewis, 1973) and four new indices based on the Tucker–Lewis index
were the only ones relatively independent of sample size. Bollen (1990)
argued that the claims regarding which model-fit indices were affected
by sample size needed further clarification. There are actually two sample
size effects that are confounded: (a) whether sample size enters into the
calculation of the model-fit index, and (b) whether the means of the sampling distribution of the model-fit index are related to sample size. Sample
size was shown not to affect the calculation of NFI, TLI, GFI, AGFI, and
CN, but the means of the sampling distribution of these model-fit indices

Y102005.indb 113

3/22/10 3:25:45 PM

114

A Beginner’s Guide to Structural Equation Modeling

were related to sample size. Bollen (1990) concluded that, given a lack of
consensus on the best measure of fit, it is prudent to report multiple measures rather than to rely on a single choice; we concur.
Muthén and Muthén (2002) also used Mplus to determine appropriate
sample sizes in the presence of model complexity, distribution of variables, missing data, reliability, and variance–covariance of variables. For
example, given a two-factor CFA model and 10 indicator variables with
normally distributed nonmissing data, a sample size of 150 is indicated
with power = .81. In the presence of missing data, sample size increases to
n = 175. Given nonnormal missing data, sample size increases to n = 315.
Davey & Savla (2009) provide an excellent treatment of statistical power
analysis with missing data via a structural equation modeling approach.
Their examples cover many different types of modeling situations using
SAS, STATA, SPSS, or LISREL syntax programs. This is a must-read book
on the subject of power and sample size, especially in the presence of
missing data.
Finally, one should beware of claims of sample size influence on fit
measures that do not distinguish the type of sample size effect (Satorra &
Bentler, 1994). Cudeck and Henly (1991) also argued that a uniformly negative view of the effects of sample size in model selection is unwarranted.
They focused instead on the predictive validity of models in the sense of
cross-validation in future samples while acknowledging that sample size
issues are a problem in the field of statistics in general and unavoidable in
structural equation modeling.

5.7 Two-Step Versus Four-Step Approach to Modeling
Anderson and Gerbing (1988) proposed a two-step model-building
approach that emphasized the analysis of two conceptually distinct models: a measurement model followed by the structural model (Lomax, 1982).
The measurement model, or factor model, specifies the relationships among
measured (observed) variables underlying the latent variables. The structural model specifies relationships among the latent variables as posited
by theory. The measurement model provides an assessment of convergent
and discriminant validity, and the structural model provides an assessment of nomological validity.
Mulaik et al. (1989) expanded the idea of model fit by assessing the relative fit of the structural model among latent variables, independently of
assessing the fit of the indicator variables in the measurement model. The
relative normed fit index (RNFI) makes the following adjustment to separately estimate the effects of the structural model from the measurement

Y102005.indb 114

3/22/10 3:25:45 PM

Model Fit

115

model: RNFIj = (Fu − Fj)/[Fu − Fm − (dfj − dfm)], where Fu = c2 of the full model,
Fj = c2 of the structural model, Fm = c2 of the measurement model, dfj is
the degrees of freedom for the structural model, and dfm is the degrees
of freedom for the measurement model. A corresponding relative parsimony ratio (RP) is given by RPj = (dfj − dfm)/(dfu − dfm), where dfj is the
degrees of freedom for the structural model, dfm is the degrees of freedom
for the measurement model, and dfu is the degrees of freedom for the null
model. In comparing different models for fit, Mulaik et al. multiplied RPj
by RNFIj to obtain a relative parsimony fit index appropriate for assessing how well and to what degree the models explained both relationships
in the measurement of latent variables and the structural relationships
among the latent variables by themselves. McDonald and Marsh (1990),
however, doubted whether model parsimony and goodness of fit could be
captured by this multiplicative form because it is not a monotonic increasing function of model complexity. Obviously, further research will be
needed to clarify these issues.
Mulaik and Millsap (2000) also presented a four-step approach to testing a nested sequence of SEM models:
• Step 1 pertains to specifying an unrestricted measurement
model, namely conducting an exploratory common factor analysis to determine the number of factors (latent variables) that fit the
variance–covariance matrix of the observed variables.
• Step 2 involves a confirmatory factor analysis model that tests
hypotheses about certain relations among indicator variables and
latent variables. Basically, certain factor loadings are fixed to zero
in an attempt to have only a single nonzero factor loading for each
indicator variable of a latent variable. Sometimes this leads to a
lack of measurement model fit because an indicator variable may
have a relation with another latent variable.
• Step 3 involves specifying relations among the latent variables in a
structural model. Certain relations among the latent variables are fixed
to zero so that some latent variables are not related to one another.
• Step 4 continues if an acceptable fit of the structural model is
achieved, that is, CFI > .95 and RMSEA < .05. In Step 4, a researcher
tests planned hypotheses about free parameters in the model.
Several approaches are possible: (a) perform simultaneous tests
in which free parameters are fixed based on theory or estimates
obtained from other research studies; (b) impose fixed parameter
values on freed parameters in a nested sequence of models until
a misspecified model is achieved (misspecified parameter); or (c)
perform a sequence of confidence-interval tests around free parameters using the standard errors of the estimated parameters.

Y102005.indb 115

3/22/10 3:25:45 PM

116

A Beginner’s Guide to Structural Equation Modeling

We agree with the basic Mulaik and Millsap (2000) approach and recommend that the measurement models for latent variables be established
first and then structural models establishing relationships among the
latent independent and dependent variables be formed. It is in the formulation of measurement models that most of the model modifications occur
to obtain acceptable data to model fit. In fact, a researcher could begin
model generation by using exploratory factor analysis (EFA) on a sample of data to find the number and type of latent variables in a plausible
model (Costello & Osborne, 2005). Once a plausible model is determined,
another sample of data could be used to confirm or test the factor model,
that is, confirmatory factor analysis (CFA) (Jöreskog, 1969). Exploratory
factor analysis is even recommended as a precursor to confirmatory factor analysis when the researcher does not have a substantive theoretical
model (Gerbing & Hamilton, 1996).
Measurement invariance is also important to examine, which refers
to considering similar measurement models across different groups; for
example, does the factor (latent variable) imply the same thing to boys
and girls? This usually involves adding between group constraints in the
measurement model. If measurement invariance cannot be established,
then the finding of a between group difference is questionable (Cheung
& Rensvold, 2002). Cheung and Rensvold (2002) also recommend that the
comparative fit index (CFI), gamma hat, and McDonald’s noncentrality
index (NCI) be used for testing between group measurement invariance
of CFA models rather than the goodness-of-fit index (GFI) or the likelihood ratio test (LR), also known as the chi-square difference test. Byrne
and Watkins (2003) questioned whether measurement invariance could
be established given that individual items on an instrument could exhibit
invariance or group differences. Later, Byrne and Sunita (2006) provided a
step-by-step approach for examining measurement invariance.

5.8 Summary
In this chapter, we began by discussing three approaches a researcher
could take in structural equation modeling: confirmatory models, alternative models, and model generation. We then considered categories of
model-fit indices—namely, model-fit, model comparison, and model
parsimony. In addition, current and new innovative approaches to specification searches were mentioned for the assessment of model fit in structural equation modeling. We examined in detail the different categories
of model-fit criteria because different fit indices have been developed
depending on the type of specified model tested. Generally, no single

Y102005.indb 116

3/22/10 3:25:45 PM

117

Model Fit

model-fit index is sufficient for testing a hypothesized structural model.
An ideal fit index just does not exist. This is not surprising because it has
been suggested that an ideal fit index is one that is independent of sample
size, accurately reflects differences in fit, imposes a penalty for inclusion
of additional parameters (Marsh et al., 1988), and supports the choice of
a true model when it is known (McDonald & Marsh, 1990). The current
model fitting practice in LISREL involves the use of modification indices
and/or expected parameter change values, but other advances in specification search techniques have been investigated (Tabu and optimization
algorithms), with a specification search approach already in AMOS (SPSS,
2009).
A two-factor confirmatory model was analyzed using the LISREL computer program with model-fit output to enhance our understanding of
the many different model-fit criteria. We concluded in this chapter with
a discussion of a four-step approach to SEM modeling, the significance
of parameters in a model, power, and sample size. An understanding of
model-fit criteria, power, and sample size will help your understanding of
the examples presented in the remaining chapters of the book.
Exercises
1. Define confirmatory models, alternative models, and modelgenerating approaches.
2. Define model fit, model comparison, and model parsimony.
3. Calculate the following fit indices for the model output in
Figure 5.1:
GFI = 1 – (c2model/c2null)
NFI = (c2null − c2model)/c2null
RFI = 1 – [(c2model/dfmodel)/(c2null/dfnull)]
IFI = (c2null − c2model)/(c2null − dfmodel)
TLI = [(c2null/dfnull) − (c2model/dfmodel)]/[(c2null/dfnull) − 1]
CFI = 1 – [(c2model − dfmodel)/(c2null − dfnull)]
Model AIC = c2model + 2q (q is the number of free parameters)
Null AIC = c2 null + 2q (q is the number of free parameters)
2
RMSEA = [ χ Model
− dfModel ]/[( N − 1)dfModel ]

RMSEA = ( NCP/N − 1)/df
4. How are modification indices in LISREL--SIMPLIS used?
5. What steps should a researcher take in examining parameter
estimates in a model?
6. How should a researcher test for the difference between two
alternative models?

Y102005.indb 117

3/22/10 3:25:46 PM

118

A Beginner’s Guide to Structural Equation Modeling

7. How are structural equation models affected by sample size
and power considerations?
8. Describe the four-step approach for modeling in SEM.
9. What new approaches are available to help a researcher identify the best model?
10. Use G*Power 3 to calculate power for modified model with
NCP = 6.3496 at p = .05, p = .01, and p = .001 levels of significance.
What happens to power when alpha increases?
11. Use G*Power 3 to calculate power for modified model with
alpha = .05 and NCP = 6.3496 at df = 1, df = 2, and df = 3 levels
of model complexity. What happens to power when degrees of
freedom increases?

Chapter Footnote
LISREL computes two different sets of standard errors for parameter estimates and up to four different chi-squares for testing overall fit of the
model. These new standard errors and chi-squares can be obtained for
single-group problems as well as multiple-group problems using variance–
covariance matrices with or without means.
Which standard errors and which chi-squares will be reported
depends on whether an asymptotic covariance matrix is provided and
which method of estimation is used to fit the model (ULS, GLS, ML,
WLS, DWLS). The asymptotic covariance matrix is a consistent estimate of N times the asymptotic covariance matrix of the sample matrix
being analyzed.

Standard Errors
Standard errors are estimated under nonnormality if an asymptotic covariance matrix is used. Standard errors are estimated under multivariate
normality if no asymptotic covariance matrix is used.

Chi-Squares
Four different chi-squares are reported and denoted below as C1, C2, C3,
and C4, where the x indicates that it is reported for any of the five estimation methods.

Y102005.indb 118

3/22/10 3:25:46 PM

119

Model Fit

Asymptotic covariance matrix not provided:
C1
C2
C3
C4

ULS
—
×
—
—

GLS
×
×
—
—

ML
×
×
—
—

WLS
—
—
—
—

DWLS
—
—
—
—

ML
×
×
×
×

WLS
×
—
—
—

DWLS
—

Asymptotic covariance matrix provided:

C1
C2
C3
C4

ULS
—
×
×
×

GLS
×
×
×
×

×
×
×

1. C1 is n − 1 times the minimum value of the fit function; C2 is
n − 1 times the minimum of the WLS fit function using a weight matrix
estimated under multivariate normality; C3 is the Satorra–Bentler scaled
chi-square statistic or its generalization to mean and covariance structures and multiple groups (Satorra & Bentler, 1994); C4 is computed from
equations in Browne (1984) or Satorra (1993) using the asymptotic covariance matrix.

Note :

The corresponding chi-squares are now given in the output as follows:
C1: Minimum fit function chi-square
C2: Normal theory weighted least squares chi-square
C3: Satorra-Bentler scaled chi-square
C4: Chi-square corrected for nonnormality
Under multivariate normality of the observed variables, C1 and
C2 are asymptotically equivalent and have an asymptotic chi-square
distribution if the model holds exactly and an asymptotic noncentral
chi-square distribution if the model holds approximately. Under normality and nonnormality, C2 and C4 are correct asymptotic chi-squares,
but may not be the best chi-square in small and moderate samples. Hu,
Bentler, and Kano (1992) and Yuan and Bentler (1997) found that C3 performed better given different types of models, sample size, and degrees
of nonnormality.

Note 2:

Y102005.indb 119

3/22/10 3:25:46 PM

120

A Beginner’s Guide to Structural Equation Modeling

References
Akaike, H. (1987). Factor analysis and AIC. Psychometrika, 52, 317–332.
Anderson, J. C., & Gerbing, D. W. (1984). The effects of sampling error on convergence, improper solutions and goodness-of-fit indices for maximum likelihood confirmatory factor analysis. Psychometrika, 49, 155–173.
Anderson, J. C., & Gerbing, D. W. (1988). Structural equation modeling in practice:
A review and recommended two-step approach. Psychological Bulletin, 103,
411–423.
Baldwin, B. (1989). A primer in the use and interpretation of structural equation models. Measurement and Evaluation in Counseling and Development, 22,
100–112.
Bentler, P. M. (1990). Comparative fit indexes in structural models. Psychological
Bulletin, 107, 238–246.
Bentler, P. M., & Bonett, D. G. (1980). Significance tests and goodness-of-fit in the
analysis of covariance structures. Psychological Bulletin, 88, 588–606.
Bollen, K. A. (1989). Structural equations with latent variables. New York: Wiley.
Bollen, K. A. (1990). Overall fit in covariance structure models: Two types of sample size effects. Psychological Bulletin, 107, 256–259.
Bollen, K. A., & Liang, J. (1988). Some properties of Hoelter’s CN. Sociological
Methods and Research, 16, 492–503.
Bollen, K. A., & Long, S. J. (1993). Testing structural equation models. Newbury Park,
CA: Sage.
Browne, M. W. (1984). Asymptotically distribution-free methods for the analysis of
covariance structures. British Journal of Mathematical and Statistical Psychology,
37, 62–83.
Browne, M. W., & Cudeck, R. (1993). Alternative ways of assessing model fit. In K.
A. Bollen & J. S. Long (Eds.), Testing structural equation models (pp. 132–162).
Beverly Hills, CA: Sage.
Byrne, B. M., & Watkins, D. (2003). The issue of measurement invariance revisited.
Journal of Cross-Cultural Psychology, 34(2), 155–175.
Byrne, B., & Sunita, M. S. (2006). The MACS approach to testing for multigroup
invariance of a second-order structure-A walk through the process. Structural
Equation Modeling: A Multidisciplinary Journal, 13(2), 287–321.
Cheung, G. W., & Rensvold, R. B. (2002). Evaluating goodness-of-fit indexes for
testing measurement invariance. Structural Equation Modeling, 9, 233–255.
Cliff, N. (1983). Some cautions concerning the application of causal modeling
methods. Multivariate Behavioral Research, 18, 115–126.
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.).
Hillsdale, NJ: Lawrence Erlbaum.
Costello, A. B., & Osborne, J. (2005). Best practices in exploratory factor analysis: Four recommendations for getting the most from your analysis. Practical
Assessment Research & Evaluation, 10(7), 1–9.
Cudeck, R., & Henly, S. J. (1991). Model selection in covariance structure analysis
and the “problem” of sample size: A clarification. Psychological Bulletin, 109,
512–519.

Y102005.indb 120

3/22/10 3:25:46 PM

Model Fit

121

Davey, A., & Savla, J. (2010). Statistical Power analysis with missing data: A structural
equation modeling approach. Routledge: Taylor & Francis, New York.
Faul, F., Erdfelder, E., Lang, A.-G., & Buchner, A. (2007). G*Power 3: A flexible
statistical power analysis program for the social, behavioral, and biomedical
sciences. Behavior Research Methods, 39, 175–191.
Gerbing, D. W., & Hamilton, J. G. (1996). Viability of exploratory factor analysis
as a precursor to confirmatory factor analysis, Structural Equation Modeling,
3(1), 62–72.
Gonzalez, R., & Griffin, D. (2001). Testing parameters in structural equation modeling: Every “one” matters. Psychological Methods, 6(3), 258–269.
Gorsuch, R. L. (1983). Factor analysis (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.
Hair, J. F., Jr., Anderson, R. E., Tatham, R. L., & Black, W. C. (1992). Multivariate data
analysis with readings (3rd ed.). New York: Macmillan.
Harmon, H. H. (1976). Modern factor analysis (3rd ed.). Chicago, IL: University of
Chicago Press.
Hoelter, J. W. (1983). The analysis of covariance structures: Goodness-of-fit indices.
Sociological Methods and Research, 11, 325–344.
Holzinger, K. J., & Swineford, F. A. (1939). A study in factor analysis: The stability of a
bi-factor solution. Supplementary Educational Monographs, No. 48. Chicago:
University of Chicago, Dept. of Education.
Hu, L., & Bentler, P. M. (1995). Evaluating model fit. In R. H. Hoyle (Ed.), Structural
equation modeling: Concepts, issues, and applications (pp. 76–99). Thousand
Oaks, CA: Sage.
Hu, L., Bentler, P. M., & Kano, Y. (1992). Can test statistics in covariance structure
analysis be trusted? Psychological Bulletin, 112, 351–362.
James, L. R., Mulaik, S. A., & Brett, J. M. (1982). Causal analysis: Assumptions, models,
and data. Beverly Hills, CA: Sage.
Jöreskog, K. G. (1969). A general approach to confirmatory maximum likelihood
factor analysis. Psychometrika, 34, 183–202.
Jöreskog, K. G., & Sörbom, D. (1993). LISREL 8: Structural equation modeling with the
SIMPLIS command language. Hillsdale, NJ: Lawrence Erlbaum.
Kaplan, D. (1995). Statistical power in structural equation modeling. In R. H.
Hoyle (Ed.), Structural equation modeling: Concepts, issues, and applications (pp.
100–117). Thousand Oaks, CA: Sage.
Kenny, D. A., & McCoach, D. B. (2003). Effect of the number of variables on measures of fit in structural equation modeling. Structural Equation Modeling, 10,
333–351.
Loehlin, J. C. (1987). Latent variable models: An introduction to factor, path, and structural analysis. Hillsdale, NJ: Lawrence Erlbaum.
Lomax, R. G. (1982). A guide to LISREL-type structural equation modeling. Behavior
Research Methods and Instrumentation, 14, 1–8.
Lunneborg, C. E. (1987). Bootstrap applications for the behavioral sciences. Vol. 1.
Seattle: University of Washington, Psychology Department.
MacCallum, R. C., Browne, M. W., & Sugawara, H. M. (1996). Power analysis and
determination of sample size for covariance structure modeling. Psychological
Methods, 1, 130–149.

Y102005.indb 121

3/22/10 3:25:46 PM

122

A Beginner’s Guide to Structural Equation Modeling

MacCallum, R. C., Browne, M. W., & Cai, L. (2006). Testing differences between
nested covariance structure models: Power analysis and null hypotheses.
Psychological Methods, 11, 19–35.
Marcoulides, G. A., & Drezner, Z. (2001). Specification searches in structural
equation modeling with a genetic algorithm. In G. A. Marcoulides & R. E.
Schumacker (Eds.), New developments and techniques in structural equation modeling (pp. 247–268). Mahwah, NJ: Lawrence Erlbaum.
Marcoulides, G. A., & Drezner, Z. (2003). Model specification searches using ant
colony optimization algorithms. Structural Equation Modeling, 10, 154–164.
Marcoulides, G. A., Drezner, Z., & Schumacker, R. E. (1998). Model specification searches in structural equation modeling using Tabu search. Structural
Equation Modeling, 5, 365–376.
Marsh, H. W., Balla, J. R., & Hau, K.-T. (1996). An evaluation of incremental fit
indices: A clarification of mathematical and empirical properties. In G. A.
Marcoulides & R. E. Schumacker (Eds.), Advanced structural equation modeling: Issues and techniques (pp. 315–353). Mahwah, NJ: Lawrence Erlbaum.
Marsh, H. W., Balla, J. R., & McDonald, R. P. (1988). Goodness-of-fit indexes in
confirmatory factor analysis: The effect of sample size. Psychological Bulletin,
103, 391–410.
McDonald, R. P. (1989). An index of goodness-of-fit based on noncentrality. Journal
of Classification, 6, 97–103.
McDonald, R. P., & Marsh, H. W. (1990). Choosing a multivariate model:
Noncentrality and goodness of fit. Psychological Bulletin, 107, 247–255.
Mulaik, S. A., James, L. R., Alstine, J. V., Bennett, N., Lind, S., & Stilwell, C. D.
(1989). Evaluation of goodness-of-fit indices for structural equation models.
Psychological Bulletin, 105, 430–445.
Mulaik, S. A., & Millsap, R. E. (2000). Doing the four-step right. Structural Equation
Modeling, 7, 36–73.
Muthén, B., & Muthén, L. (2002). How to use a Monte Carlo study to decide on
sample size and determine power. Structural Equation Modeling, 9, 599–620.
Saris, W. E., & Satorra, A. (1993). Power evaluation in structural equation models. In K. Bollen & J. S. Long (Eds.), Testing structural equation models (pp.
181–204). Newbury Park, CA: Sage.
Satorra, A. (1993). Multi-sample analysis of moment structures: Asymptotic
validity of inferences based on second-order moments. In K. Haagen, D. J.
Bartholomew, & M. Deistler (Eds.), Statistical modeling and latent variables (pp.
283–298). Amsterdam: Elsevier.
Satorra, A., & Bentler, P. M. (1994). Corrections for test statistics and standard
errors in covariance structure analysis. In A. Von Eye & C. C. Clogg (Eds.),
Latent variable analysis: Applications for developmental research (pp. 399–419).
Thousand Oaks, CA: Sage.
Soper, D., Statistics Calculators. Retrieved January 2010 from http://www.danielsoper.com/statcalc/.
SPSS (2009). Statistics 17.0. SPSS, Inc.: Chicago, IL.
Steiger, J. H. (1990). Structural model evaluation and modification: An interval
estimation approach. Multivariate Behavioral Research, 25, 173–180.

Y102005.indb 122

3/22/10 3:25:46 PM

Model Fit

123

Steiger, J. H., & Lind, J. M. (1980, May). Statistically-based tests for the number of common factors. Paper presented at Psychometric Society Meeting, Iowa City, IA.
Tucker, L. R., & Lewis, C. (1973). The reliability coefficient for maximum likelihood
factor analysis. Psychometrika, 38, 1–10.
Williams, L. J., & Holahan, P. J. (1994). Parsimony-based fit indices of multiple indicator models: Do they work? Structural Equation Modeling: A Multidisciplinary
Journal, 1, 161–189.
Yuan, K.-H., & Bentler, P. M. (1997). Mean and covariance structure analysis:
Theoretical and practical improvements. Journal of the American Statistical
Association, 92, 767–774.

Y102005.indb 123

3/22/10 3:25:46 PM

6
Regression Models

Key Concepts
Explanation versus prediction
Standardized partial regression coefficients
Coefficient of determination
Squared multiple correlation coefficient
Full versus restricted models
Confidence intervals around R 2
Measurement error
Additive versus relational model
In this chapter, we consider multiple regression models as a method for
modeling multiple observed variables. Multiple regression, a general linear modeling approach to the analysis of data, has become increasingly
popular since 1967 (Bashaw & Findley, 1968). In fact, it has become recognized as an approach that bridges the gap between correlation and analysis
of variance in answering research hypotheses (McNeil, Kelly, & McNeil,
1975). Many statistical textbooks elaborate the relationship between multiple regression and analysis of variance (Draper & Smith, 1966; Edwards,
1979; Hinkle, Wiersma, & Jurs, 2003; Lomax, 2007).
Graduate students who take an advanced statistics course are typically
provided with the multiple linear regression framework for data analysis.
Given knowledge of multiple linear regression techniques (one dependent
variable), understanding can be extended to various multivariable statistical techniques (Newman, 1988). A basic knowledge of multiple regression
concepts is therefore important in further understanding path analysis as
presented in Chapter 7. This chapter shows how beta weights (standardized partial regression coefficients) are computed in multiple regression
using a structural equation modeling software program. More specifically,
we illustrate how the structural equation modeling approach can be used
125

Y102005.indb 125

3/22/10 3:25:47 PM

126

A Beginner’s Guide to Structural Equation Modeling

to compute parameter estimates in multiple regression and what model-fit
criteria are reported. We begin with a brief overview of multiple regression
concepts followed by an example that illustrates model specification, model
identification, model estimation, model testing, and model modification.

6.1 Overview
Multiple regression techniques require a basic understanding of sample
statistics (sample size, mean, and variance), standardized variables, correlation (Pedhazur, 1982), and partial correlation (Cohen & Cohen, 1983;
Houston & Bolding, 1974). In standard score form (z scores), the simple
linear regression equation for predicting the dependent variable Y from a
single independent variable X is
zˆ y = β zx ,
where b is the standardized regression coefficient. The basic rationale for
using the standard-score formula is that variables are converted to the
same scale of measurement, the z scale. Conversion back to the raw-score
scale is easily accomplished by using the raw score, the mean and the standard deviation.
The relationship connecting the Pearson product-moment correlation
coefficient, the unstandardized regression coefficient b and the standardized regression coefficient b is

β=

∑ zxzy
∑ z 2x

= b s x = r xy ,
sy

where sx and sy are the sample standard deviations for variables X and Y,
respectively. For two independent variables, the multiple linear regression
equation with standard scores is
zˆ y = β1 z1 + β 2 z 2
and the standardized partial regression coefficients b1 and b 2 are computed from

β1 =

r y 1 − r y 2 r 12
2
1 − r 12

and β 2 =

r y 2 − r y 1r 12
.
2
1 − r 12

The correlation between the dependent observed variable Y and the predicted scores Ŷ is given the special name multiple correlation coefficient. It is

Y102005.indb 126

3/22/10 3:25:48 PM

127

Regression Models

written as
R yyˆ = R y.12 ,
where the latter subscripts indicate that the dependent variable Y is being
predicted by two independent variables, X1 and X2. The squared multiple
correlation coefficient is computed as
2
2
R yyˆ = R y.12 = β 1rY +1 + β 2 rY 2 .

The squared multiple correlation coefficient indicates the amount of variance explained, predicted, or accounted for in the dependent variable by
the set of independent predictor variables. The R2 value is used as a modelfit criterion in multiple regression analysis.
Kerlinger and Pedhazur (1973) indicated that multiple regression analysis can play an important role in prediction and explanation. Prediction
and explanation reflect different research questions, study designs, inferential approaches, analysis strategies, and reported information. In prediction, the main emphasis is on practical application such that independent
variables are chosen by their effectiveness in enhancing prediction of the
dependent variable. In explanation, the main emphasis is on the variability in the dependent variable explained by a theoretically meaningful set
of independent variables. Huberty (2003) established a clear distinction
between prediction and explanation when referring to multiple correlation analysis (MCA) and multiple regression analysis (MRA). In MCA, a
parameter of interest is the correlation between the dependent variable Y
and a composite of the independent variables Xp. The adjusted formula
using sample size n and the number of independent predictors p is
2 = R2 −
RAdj

p
(1 − R 2 ) .
n− p−1

In MRA, regression weights are also estimated to achieve a composite for
the independent variables Xp, but the index of fit R 2 is computed differently as
2
RAdj*
= R2 −

2p
(1 − R 2 ).
n−p

When comparing these two formulas, we see that R 2Adj* has a larger adjustment. For example, given R 2 = .50, p = 10 predictor variables and n = 100
subjects, these two different fit indices are
2 = R2 −
RAdj

Y102005.indb 127

p
(1 − R 2 ) = .50 − .11(.50) = .50 − .055 = .45
n− p−1

3/22/10 3:25:49 PM

128

A Beginner’s Guide to Structural Equation Modeling

2
RAdj*
= R2 −

2p
(1 − R 2 ) = .50 − .22(.50) = .50 − .11 = .39 .
n− p

Hypothesis testing would involve using the expected value or chance value
of R2 for testing the null hypothesis, which is p/(n − 1), not 0 as typically
indicated. In our example, the expected or chance value for R2 = 10/99 = .10,
so the null hypothesis is H0: r 2 = .10. An F test used to test the statistical
significance of the R 2 value is
F=

R 2 /p
.
(1 − R )/n − p − 1
2

In our example,
F=

.50/10
.05
R 2 /p
= 8.9,
=
=
2
(1 − R )/n − p − 1 (1 − .50)/89 .0056

which is statistically significant when compared to the tabled F = 1.93, df =
10,89, p < .05 (Table A.5). In addition to the statistical significance test, a
researcher should calculate effect sizes and confidence intervals to aid understanding and interpretation (Soper, 2010).
The effect size (ES) is computed as ES = R2 – [p/(n − 1)]. In our example, ES
R2Adj = .45 − .10 = .35 and ES R2Adj* = .39 − .10 = .29. This indicates a moderate
to large effect size according to Cohen (1988), who gave a general reference
for effect sizes (small = .1, medium = .25, and large = .4).
Confidence intervals (CIs) around the R2 value can also help our interpretation of multiple regression analysis. Steiger and Fouladi (1992) reported
an R 2 CI DOS program that computes confidence intervals, power, and
sample size. Steiger and Fouladi (1997) and Cumming and Finch (2001)
both discussed the importance of converting the central F value to an estimate of the noncentral F before computing a confidence interval around
R 2. Smithson (2001) wrote an R2 SPSS program to compute confidence
intervals.
We use the Steiger and Fouladi (1997) R 2 CI DOS program with our hypothetical example. After entering the program, Option is selected from the
tool bar menu and then Confidence Interval is selected from the dropdown menu. To obtain R 2 CI, the number of subjects (n = 100), the number
of variables (K = 10), the R 2 value (R = .35), and the desired confidence level
(C = .95) are entered by using the arrow keys (mouse not supported), and
then GO is selected to compute the values. The 95% confidence interval
around R2 = .35 is .133 to .449 at the p = .0001 level of significance for a null
hypothesis that R 2 = 0 in the population.

Y102005.indb 128

3/22/10 3:25:50 PM

129

Regression Models

After assessing our initial regression model fit, we might want to
determine whether adding or deleting an independent variable would
improve the index of fit R2, but we avoid using stepwise regression methods (Huberty, 1989). We run a second multiple regression equation where
a single independent variable is added or deleted to obtain a second R 2
value. We then compute a different F test to determine the statistical significance between the two regression models as follows
F=

(RF2 − RR2 )/( p1 − p2 )
,
(1 − RF2 )/n − p1 − 1

where R 2F is from the multiple regression equation with the full original
set of independent variables p1 and R2R is from the multiple regression
equation with the reduced set of independent variables p2. In our heuristic
example, we drop a single independent variable and obtain R 2R = .49 with
p2 = 9 predictor variables. The F test is computed as:
F=

(RF2 − RR2 )/( p1 − p2 )
(.50 − .49)/(10 − 9)
.01
=
=
= 1.78 .
(1 − RF2 )/n − p1 − 1 (1 − .50)/100 − 10 − 1 .0056

The F value is not significant at the .05 level, so the variable we dropped
does not statistically add to the prediction of Y, which supports our dropping the single predictor variable; that is, a 1% decrease in R 2 is not statistically significant. The nine-variable regression model therefore provides a
more parsimonious model.
It is important to understand the basic concepts of multiple regression and correlation because they provide a better understanding of path
analysis in chapter 7, and structural equation modeling in general. An
example is presented next to further clarify these basic multiple regression
computations.

Y102005.indb 129

3/22/10 3:25:50 PM

130

A Beginner’s Guide to Structural Equation Modeling

6.2 An Example
A multiple linear regression analysis is conducted using data from
Chatterjee and Yilmaz (1992). The data file contains scores from 24 patients
on four variables (Var1 = patient’s age in years, Var2 = severity of illness,
Var3 = level of anxiety, and Var4 = satisfaction level). Given raw data, two
different approaches are possible in LISREL: (a) a system file in LISREL–
PRELIS using regression statistics from the pull-down menu or (b) a correlation or covariance matrix input in the LISREL–SIMPLIS command
syntax file. We choose to compute and input a covariance matrix into a
LISREL–SIMPLIS program.

6.3 Model Specification
Model specification involves finding relevant theory and prior research
to formulate a theoretical regression model. The researcher is interested
in specifying a regression model that should be confirmed with sample
variance–covariance data, thus yielding a high R 2 value and statistically significant F value. Model specification directly involves deciding
which variables to include or not to include in the theoretical regression
model.
If the researcher does not select the right variables, then the regression
model could be misspecified and lack validity (Tracz, Brown, & Kopriva,
1991). The problem is that a misspecified model may result in biased
parameter estimates or estimates that are systematically different from
what they really are in the true population model. This bias is known as
specification error.
The researcher’s goal is to determine whether the theoretical regression
model fits the sample variance–covariance structure in the data, that is,
whether the sample variance–covariance matrix implies some underlying
theoretical regression model. The multiple regression model of theoretical
interest in our example is to predict the satisfaction level of patients based
on patient’s age, severity of illness, and level of anxiety (independent variables). This would be characteristic of a MCA model because a particular
set of variables were selected based on theory. The dependent variable
Var4 is therefore predicted by the three independent variables (Var1, Var2,
and Var3). The path diagram of the implied regression model is shown in
Figure 6.1.

Y102005.indb 130

3/22/10 3:25:50 PM

131

Regression Models

var1

var2

var4

error

var3
Figure 6.1
Satisfaction regression model.

6.4 Model Identification
Once a theoretical regression model is specified, the next concern is model
identification. Model identification refers to deciding whether a set of
unique parameter estimates can be computed for the regression equation.
Algebraically, every free parameter in the multiple regression equation can
be estimated from the sample variance–covariance matrix (a free parameter
is an unknown parameter that you want to estimate). The number of distinct values in the sample variance–covariance matrix equals the number
of parameters to be estimated; thus, multiple regression models are always
considered just-identified (see chapter 4). SEM computer output will therefore indicate that regression analyses are saturated models; thus, c2 = 0 and
degrees of freedom = 0. There are 3 variances, 3 covariance terms, 3 regression
weights, and 1 error term so all parameters in the regression equation are
being estimated.

6.5 Model Estimation
Model estimation involves estimating the parameters in the regression model—
that is, computing the sample regression weights for the independent predictor variables. The squared multiple correlation with three predictor variables
(VAR1, VAR2, VAR3) predicting the dependent variable Y (VAR4) is
2
R y.123 = β 1r y 1 + β 2r y 2 + β 3r y 3 .

Y102005.indb 131

3/22/10 3:25:51 PM

132

A Beginner’s Guide to Structural Equation Modeling

The correlation coefficients are multiplied by their respective standardized partial regression weights and summed to yield the squared multiple
regression coefficient R 2y . 123.
In LISREL–SIMPLIS, we select File, New, and then Syntax Only in the
dialog box to write the following SIMPLIS program:
Regression Analysis Example (no intercept term)
Observed variables: VAR1 VAR2 VAR3 VAR4
Covariance matrix:
91.384
30.641 27.288
0.584 0.641 0.100
−122.616 −52.576 −2.399 281.210
Sample size: 24
Equation: VAR4 = VAR1 VAR2 VAR3
Number of decimals = 3
Path Diagram
End of Problem

You will be prompted to save the program with a file name (*.spl) before
the program runs.
The critical portion of the LISREL–SIMPLIS regression output without
an intercept term in the regression equation looks like:
VAR4 = − 1.153*VAR1 − 0.267*VAR2 − 15.546*VAR3, Errorvar.= 88.515, R² = 0.685
(0.279)
−4.129

(0.544)
−0.491

(7.232)

(27.991)

−2.150

3.162

Goodness-of-Fit Statistics
Degrees of Freedom = 0
Minimum Fit Function Chi-Square = 0.0 (P = 1.000)

Normal Theory Weighted Least Squares Chi-Square =
0.00 (P = 1.000)
The model is saturated, the fit is perfect!

We notice that the regression weights are identified for each independent
variable (VAR1 – VAR3). Below each regression weight is the standard
error in parenthesis, for example, VAR1 regression weight has a standard
error of .279; with the T or Z value indicated below that, and a p-value
listed below the T or Z value. (Note: LISREL 8.8 Student version lists the
parameter estimate, standerd error, z value, and associated p-value.) Recall
that T = parameter divided by standard error (T = −1.153/.279 = −4.129). If
testing each regression weight at the critical t = 1.96, a = .05 level of significance, then VAR1 and VAR3 are statistically significant, but VAR2 is
not (T = −.491). We also notice that R2 = .685 or 69% of the variability in
Y scores (VAR4) is predicted by knowledge of VAR1, VAR2, and VAR3.

Y102005.indb 132

3/22/10 3:25:51 PM

133

Regression Models

We will later discuss modifying this regression model—that is, dropping
VAR2 (see section 6.7). This example is further explained in Jöreskog and
Sörbom (1993, pp. 1–6).

6.6 Model Testing
Model testing involves determining the fit of the theoretical model.
Therefore, we will present how to hand calculate the R 2 value from the
correlation matrix output by LISREL, as follows:
CORRELATION MATRIX
VAR1
VAR1
1.0000
VAR2
0.6136
VAR3
0.1935
VAR4
−0.7649

VAR2

VAR3

VAR4

1.0000
0.3888
−0.6002

1.0000
−0.4530

1.0000

The standardized regression coefficients can be obtained from selecting
the standardized solution in the pull down menu of the path diagram window of the LISREL–SIMPLIS program. We can now verify the R 2 value
using the standardized regression formula:
This matches the R 2 value in the LISREL–SIMPLIS output as shown
above). The adjusted R 2 value for the MCA theoretical regression model
approach is
(Note:

R y.123 = β 1r y 1 + β 2r y 2 + β 3r y 3 = −.657(−.7649) + −.083(−.6002)
+ −.294(−.4530) = .685.
2 = R2 −
RAdj

p
(1 − R 2 ) = .685 − .15(.315) = .685 − .0047 = .638.
n− p−1

The F test for the significance of the R 2 value is
F=

.685/3
.228
R 2 /p
= 14.52 .
=
=
(1 − R )/n − p − 1 (1 − .685)/20 .0157
2

The effect size is
R2 – [p/(n − 1)] = .685 – [3/23] = .685 − .130 = .554.

Y102005.indb 133

3/22/10 3:25:52 PM

134

A Beginner’s Guide to Structural Equation Modeling

This is a large effect size. The 95% confidence interval around R 2 = .685 using
the R2 CI program is (.33, .83).
The results indicate that a patient’s age, severity of illness, and level of
anxiety make up a statistically significant set of predictors of a patient’s
satisfaction level. There is a large effect size and the confidence interval
reveals the range of R2 values one can expect in conducting a regression
analysis on another sample of data. The negative standardized regression
coefficients indicate that as patient age, severity of illness, and anxiety
increase, patient satisfaction decreases.

6.7 Model Modification
The theoretical regression model included a set of three independent explanatory variables, which resulted in a statistically significant R2 = .685. This
implies that 69% of the patient satisfaction level score variance is explained
by knowledge of a patient’s age, severity of illness, and level of anxiety. The
regression analysis, however, indicated that the regression weight for Var2
was not statistically different from zero (t = −0.491, p > .10). Thus, one might
consider model modification where the theoretical regression model is modified to produce a two-variable regression equation, thus allowing for the
F test of the difference between the two regression analysis R2 values.
We repeat the steps for the regression analysis, but this time only
including Var1 and Var3 in the analysis. The results for the regression
equation with these two variables, Var1 and Var3 in the LISREL–SIMPLIS
program, are:
VAR4 = − 1.235*VAR1 − 16.780*VAR3, Errorvar. = 89.581, R² = 0.681
(0.220)
−5.606

(6.657)
−2.521

(27.645)
3.240

The F test for a difference between the two models is
F=

(RF2 − RR2 )/( p1 − p2 ) (.685 − .681)/(3 − 2) .004
=
= .25 .
=
(1 − RF2 )/n − p1 − 1 (1 − .685)/24 − 3 − 1 .016

The F test for the difference in the two R 2 values was nonsignificant indicating that dropping Var2 does not affect the explanation of a patient’s
satisfaction level (R2 = .685 vs. R2 = .681). We therefore use the more parsimonious two-variable regression model (68% of the variance in a patient’s
satisfaction level is explained by knowledge of a patient’s age and level of
anxiety, that is, 68% of 281.210 = 191.22).

Y102005.indb 134

3/22/10 3:25:52 PM

Regression Models

135

Because the R2 value is not 1.0 (perfect explanation or prediction), additional variables could be added if more recent research indicated that
another variable was relevant to a patient’s satisfaction level, for example,
the number of psychological assessment visits. Obviously, more variables
can be added in the model modification process, but a theoretical basis
should be established by the researcher for the additional variables.

6.8 Summary
This chapter illustrated the important statistics to report when conducting
a regression analysis. We found that the model-fit statistics in chapter 5 do
not apply because regression models are saturated just-identified models.
We also showed that the selection of independent variables in the regression model (model specification) and the subsequent regression model
modification are key issues not easily resolved without a good sound
theoretical justification.
The selection of a set of independent variables and the subsequent
regression model modification are important issues in multiple regression.
How does a researcher determine the best set of independent variables
for explanation or prediction? It is highly recommended that a regression
model be based on some theoretical framework that can be used to guide
the decision of what variables to include. Model specification consists of
determining what variables to include in the model and which variables
are independent or dependent. A systematic determination of the most
important set of variables can then be accomplished by setting the partial regression weight of a single variable to zero, thus testing full and
restricted models for a difference in the R2 values (F test). This approach
and other alternative methods were presented by Darlington (1968).
In multiple regression, the selection of a wrong set of variables can yield
erroneous and inflated R2 values. The process of determining which set
of variables yields the best prediction, given time, cost, and staffing, is
often problematic because several methods and criteria are available to
choose from. Recent methodological reviews have indicated that stepwise
methods are not preferred, and that an all-possible-subset approach is recommended (Huberty, 1989; Thompson, Smith, Miller, & Thomson, 1991).
In addition, the Mallows CP statistic is advocated by some rather than R2
for selecting the best set of predictors (Mallows, 1966; Schumacker, 1994;
Zuccaro, 1992). Overall, which variables are included in a regression equation will determine the validity of the model and be determined by the
rationale for the model by the researcher (see Chapter Note, for inclusion
of an intercept term).

Y102005.indb 135

3/22/10 3:25:52 PM

136

A Beginner’s Guide to Structural Equation Modeling

Because multiple regression techniques have been shown to be robust
to violations of assumptions (Bohrnstedt & Carter, 1971) and applicable to
contrast coding, dichotomous coding, ordinal coding (Lyons, 1971), and
criterion scaling (Schumacker, 1993), they have been used in a variety of
research designs. In fact, multiple regression equations can be used to
address several different types of research questions. The model specification issue, however, is paramount in achieving a valid multiple regression model. Replication, cross-validation, and bootstrapping have all been
applied in multiple regression to determine the validity of a regression
model (see chapter 12 for further discussion of these methods in SEM).
There are other issues related to using the regression method, namely,
variable measurement error and the additive nature of the equation. These
two issues are described next.
6.8.1 Measurement Error
The issue of unreliable variable measurements and their effect on multiple regression has been previously discussed (Cleary, 1969; Cochran,
1968; Fuller & Hidiroglou, 1978; Subkoviak & Levin, 1977; Sutcliffe, 1958).
A recommended solution was to multiply the dependent variable reliability and/or average of the independent variable reliabilities by the R 2 value
(Cochran, 1968, 1970). The basic equation using only the reliability of the
dependent variable is
2
2
Rˆ y.123 = R y.123 * r yy ,

or, including the dependent variable reliability and the average of the
independent variable reliabilities,
2
2
Rˆ y.123 = R y.123 * r yy * r xx .

This is not always possible if reliabilities of the dependent and independent variables are unknown. This correction to R 2 for measurement error
(unreliability) has intuitive appeal given the definition of classical reliability, namely the proportion of true score variance accounted for given
the observed scores. In our previous example, R2 = .68. If the dependent
variable reliability is .80, then only 54% of the variance in patient’s satisfaction level is true variance, rather than 68%. Similarly, if the average of
the two independent variable reliabilities was .90, then multiplying .68 by
.80 by .90 yields only 49% variance as true variance. Obviously, unreliable
variables (measurement error) can have a dramatic effect on statistics and
our interpretation of the results. Werts, Rock, Linn, and Jöreskog (1976)
examined correlations, variances, covariances, and regression weights

Y102005.indb 136

3/22/10 3:25:53 PM

Regression Models

137

with and without measurement error and developed a program to correct
the regression weights for attenuation. Our basic concern is that unreliable measured variables coupled with a potential misspecified model do
not represent theory well.
The impact of measurement error on statistical analyses is not new,
but is often forgotten by researchers. Fuller (1987) extensively covered
structural equation modeling, and especially extended regression analysis to the case where the variables were measured with error. Cochran
(1968) studied four different aspects of how measurement error affected
statistics: (a) types of mathematical models, (b) standard techniques of
analysis that take measurement error into account, (c) effect of errors of
measurement in producing bias and reduced precision and what remedial
procedures are available, and (d) techniques for studying error of measurement. Cochran (1970) also studied the effects of measurement error
on the squared multiple correlation coefficient.
The validity and reliability issues in measurement have traditionally
been handled by first examining the validity and reliability of scores on
instruments used in a particular research design. Given an acceptable
level of score validity and reliability, the scores are then used in a statistical analysis. The traditional statistical analysis of these scores using
multiple regression, however, did not adjust for measurement error,
so it is not surprising that an approach such as SEM was developed
to incorporate measurement error adjustments into statistical analyses
(Loehlin, 1992).
6.8.2 Additive Equation
The multiple regression equation is by definition additive (Y = X1 + X2)
and thus does not permit any other relationships among the variables
to be specified. This limits the potential for variables to have direct,
indirect, and total effects on each other as described in chapter 7 (path
models). In fact, a researcher’s interest should not be with the Pearson
product-moment correlations, but rather with partial or part correlations that reflect the unique additive contribution of each variable, that
is, standardized partial regression weights. Even with this emphasis,
the basic problem is that variables are typically added in a regression
model, a process that functions ideally only if all independent variables are highly correlated with the dependent variable and uncorrelated among themselves. Path models, in contrast, provide theoretically
meaningful relationships in a manner not restricted to an additive
model (Schumacker, 1991).
Multiple regression as a general data-analytic technique is widely
accepted and used by educational researchers, behavioral scientists, and
biostatisticians. Multiple regression methods basically determine the overall

Y102005.indb 137

3/22/10 3:25:53 PM

138

A Beginner’s Guide to Structural Equation Modeling

contribution of a set of observed variables to explanation or prediction,
test full and restricted models for the significant contribution of a variable
in a model, and delineate the best subset of multiple independent predictors. Multiple regression equations also permit the use of nominal, ordinal,
effect, contrast, or polynomial coded variables (Pedhazur, 1982; Pedhazur &
Schmelkin, 1992). The multiple regression approach, however, is not robust
to measurement error and model misspecification (Bohrnstedt & Carter,
1971) and gives an additive model rather than a relational model; hence,
path models play an important role in defining more meaningful theoretical
models to test.

Chapter Footnote
Regression Model with Intercept Term
In the LISREL–SIMPLIS GUIDE (Jöreskog & Sörbom, 1993) we see
our first use of the CONST command which uses a mean value, thus
includes an intercept term in the model. The SEM modeling type structured means makes use of this command to test the mean values between
models (see Chapter 13). The following LISREL–SIMPLIS Program
includes the command, CONST, to produce an intercept term in the
regression equation:

LISREL–SIMPLIS Program (Intercept Term)
Regression Analysis
Raw Data from file chatter.psf
Equation: VAR4 = VAR1 VAR2 VAR3 CONST
Path Diagram
End of Problem
The LISREL–SIMPLIS output would look like this:
VAR4 = 156.62 − 1.15*VAR1 − 0.27*VAR2 − 15.59*VAR3, Errorvar. = 88.46, R² = 0.69
(22.61)
6.93

Y102005.indb 138

(0.28)
−4.13

(0.54)
−0.49

(7.24)
−2.15

(27.97)
3.16

3/22/10 3:25:53 PM

139

Regression Models

In the LISREL 8.8 Student Examples folder, SPLEX, the program EX1A.
SPL computes the regression equation without an intercept term, while
the program EX1B.SPL computes the regression equation with an intercept term. In general, if you include sample means, then an intercept term
is included in the equation. These examples are further explained in the
LISREL8: Structural Equation Modeling with the SIMPLIS Command Language
(Jöreskog & Sörbom, 1993, p. 1–6).
Exercises
1. Analyze the regression model in LISREL–SIMPLIS using the
covariance matrix below with a sample size of 23 as described
in Jöreskog and Sörbom (1993, pp. 3–6). The theoretical regression model specifies that the dependent variable, gross national
product (GNP), is predicted by labor, capital, and time (three
independent variables).
Covariance Matrix
GNP
Labor
Capital
Time

4256.530
449.016
1535.097
537.482

52.984
139.449
53.291

1114.447
170.024

73.747

2. Is there an alternative regression model that predicts GNP
better? Report the F, effect size, and confidence interval for the
revised model. The regression model is shown in Figure 6.2
Labor

Capitol

GNP

error

Time
Figure 6.2
GNP regression model.

References
Bashaw, W. L., & Findley, W. G. (1968). Symposium on general linear model approach
to the analysis of experimental data in educational research. (Project No. 7‑8096).
Washington, DC: U.S. Department of Health, Education, and Welfare.

Y102005.indb 139

3/22/10 3:25:54 PM

140

A Beginner’s Guide to Structural Equation Modeling

Bohrnstedt, G. W., & Carter, T. M. (1971). Robustness in regression analysis. In
H. L. Costner (Ed.), Sociological methodology (pp. 118–146). San Francisco,
CA: Jossey‑Bass.
Chatterjee, S., & Yilmaz, M. (1992). A review of regression diagnostics for behavioral research. Applied Psychological Measurement, 16, 209–227.
Cleary, T. A. (1969). Error of measurement and the power of a statistical test. British
Journal of Mathematical and Statistical Psychology, 22, 49–55.
Cochran, W. G. (1968). Errors of measurement in statistics. Technometrics, 10,
637–666.
Cochran, W. G. (1970). Some effects of errors of measurement on multiple correlation. Journal of the American Statistical Association, 65, 22–34.
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.).
Hillsdale, NJ: Lawrence Erlbaum.
Cohen, J., & Cohen, P. (1983). Applied multiple regression/correlation analysis for the
behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.
Cumming, G., & Finch, S. (2001). A primer on the understanding, use and calculation of confidence intervals that are based on central and noncentral distributions. Educational and Psychological Measurement, 61, 532–574.
Darlington, R. B. (1968). Multiple regression in psychological research and practice. Psychological Bulletin, 69, 161–182.
Draper, N. R., & Smith, H. (1966). Applied regression analysis. New York: Wiley.
Edwards, A. L. (1979). Multiple regression and the analysis of variance and covariance.
San Francisco, CA: Freeman.
Fuller, W. A. (1987). Measurement error models. New York: Wiley.
Fuller, W. A., & Hidiroglou, M. A. (1978). Regression estimates after correcting for
attenuation. Journal of the American Statistical Association, 73, 99–104.
Hinkle, D. E., Wiersma, W., & Jurs, S.G. (2003). Applied statistics for the behavioral
sciences (5th ed.). Boston, MA: Houghton Mifflin.
Houston, S. R., & Bolding, J. T., Jr. (1974). Part, partial, and multiple correlation
in commonality analysis of multiple regression models. Multiple Linear
Regression Viewpoints, 5, 36–40.
Huberty, C. J. (1989). Problems with stepwise methods—Better alternatives. In
B. Thompson (Ed.), Advances in social science methodology (Vol. 1, pp. 43–70).
Greenwich, CT: JAI.
Huberty, C. J. (2003). Multiple correlation versus multiple regression. Educational
and Psychological Measurement, 63, 271–278.
Jöreskog, K. G., & Sörbom, D. (1993). LISREL8: Structural equation modeling with the
SIMPLIS command language. Chicago, IL: Scientific Software International.
Kerlinger, F. N., & Pedhazur, E. J. (1973). Multiple regression in behavioral research.
New York: Holt, Rinehart, & Winston.
Loehlin, J. C. (1992). Latent variable models: An introduction to factor, path, and structural analysis (2nd ed.). Mahwah, NJ: Lawrence Erlbaum.
Lomax, R. G. (2007). Statistical concepts: A second course (3rd ed.). Mahwah, NJ:
Lawrence Erlbaum.
Lyons, M. (1971). Techniques for using ordinal measures in regression and path
analysis. In H. L. Costner (Ed.), Sociological methodology (pp. 147–171). San
Francisco, CA: Jossey‑Bass.

Y102005.indb 140

3/22/10 3:25:54 PM

Regression Models

141

Mallows, C. L. (1966, March). Choosing a subset regression. Paper presented at the
Joint Meetings of the American Statistical Association, Los Angeles.
McNeil, K. A., Kelly, F. J., & McNeil, J. T. (1975). Testing research hypotheses using
multiple linear regression. Carbondale: Southern Illinois University Press.
Newman, I. (1988, October). There is no such thing as multivariate analysis: All analyses are univariate. President’s address at Mid‑Western Educational Research
Association, Chicago.
Pedhazur, E. J. (1982). Multiple regression in behavioral research: Explanation and prediction (2nd ed.). New York: Holt, Rinehart, & Winston.
Pedhazur, E. J., & Schmelkin, L. P. (1992). Measurement, design, and analysis: An
integrated approach. Hillsdale, NJ: Lawrence Erlbaum.
Schumacker, R. E. (1991). Relationship between multiple regression, path, factor,
and LISREL analyses. Multiple Linear Regression Viewpoints, 18, 28–46.
Schumacker, R. E. (1993). Teaching ordinal and criterion scaling in multiple regression. Multiple Linear Regression Viewpoints, 20, 25–31.
Schumacker, R. E. (1994). A comparison of the Mallows Cp and principal component regression criteria for best model selection. Multiple Linear Regression
Viewpoints, 21, 12–22.
Smithson, M. (2001). Correct confidence intervals for various regression effect
sizes and parameters: The importance of noncentral distributions in computing intervals. Educational and Psychological Measurement, 61, 605–632.
Soper, D. Statistics Calculators. Retrieved January 2010 from http://www.danielsoper.com/statcalc/.
Steiger, J. H., & Fouladi, T. (1992). R2: A computer program for interval estimation, power calculation, and hypothesis testing for the squared multiple correlation. Behavior Research Methods, Instruments, and Computers, 4,
581–582.
Steiger, J. H., & Fouladi, T. (1997). Noncentrality interval estimation and the evaluation
of statistical models. In L. Harlow, S. Mulaik, & J.H. Steiger (Eds.), What if there
were no significance tests? (pp. 222–257). Mahwah, NJ: Lawrence Erlbaum.
Subkoviak, M. J., & Levin, J. R. (1977). Fallibility of measurement and the power of
a statistical test. Journal of Educational Measurement, 14, 47–52.
Sutcliffe, J. P. (1958). Error of measurement and the sensitivity of a test of significance. Psychometrika, 23, 9–17.
Thompson, B., Smith, Q. W., Miller, L. M., & Thomson, W. A. (1991, January).
Stepwise methods lead to bad interpretations: Better alternatives. Paper presented
at the annual meeting of the Southwest Educational Research Association,
San Antonio, TX.
Tracz, S. M., Brown, R., & Kopriva, R. (1991). Considerations, issues, and comparisons in variable selection and interpretation in multiple regression. Multiple
Linear Regression Viewpoints, 18, 55–66.
Werts, C. E., Rock, D. A., Linn, R. L., & Jöreskog, K. G. (1976). Comparison of
correlations, variances, covariances, and regression weights with or without
measurement error. Psychological Bulletin, 83, 1007–1013.
Zuccaro, C. (1992). Mallows Cp statistic and model selection in multiple linear
regression. Journal of the Market Research Society, 34, 163–172.

Y102005.indb 141

3/22/10 3:25:54 PM

7
Path Models

Key Concepts
Path model diagrams
Direct effects, indirect effects, and correlated independent variables
Path (structure) coefficients and standardized partial regression
coefficients
Decomposition of correlations
Original and reproduced correlation coefficients
Full versus limited information function
Residual and standardized residual matrix
In this chapter we consider path models, the logical extension of multiple
regression models. Although path analysis still uses models involving
multiple observed variables, there may be any number of independent
and dependent variables and any number of equations. Thus, as we shall
see, path models require the analysis of several multiple regression equations using observed variables.
Sewall Wright is credited with the development of path analysis as a
method for studying the direct and indirect effects of variables (Wright,
1921, 1934, 1960). Path analysis is not actually a method for discovering
causes; rather, it tests theoretical relationships, which historically has been
termed causal modeling. A specified path model might actually establish
causal relationships among two variables when:
1.
2.
3.
4.

Temporal ordering of variables exists.
Covariation or correlation is present among variables.
Other causes are controlled for.
A variable X is manipulated, which causes a change in Y.

Obviously, a theoretical model that is tested over time (longitudinal research)
and manipulates certain variables to assess the change in other variables
143

Y102005.indb 143

3/22/10 3:25:54 PM

144

A Beginner’s Guide to Structural Equation Modeling

(experimental research) more closely approaches our idea of causation. In
the social and behavioral sciences, the issue of causation is not as straightforward as in the hard sciences, but it has the potential to be modeled.
Pearl (2009) has renewed a discussion of causation in the behavioral
sciences with model examples and rationale for causation as a process
(model) that can be expressed in mathematical expressions ready for computer analysis, which fits into the testing of theoretical path models.
This chapter begins with an example path model, and then proceeds
with sections on model specification, model identification, model estimation, model testing, and model modification.

7.1 An Example
We begin with a path model that will be followed throughout the chapter.
McDonald and Clelland (1984) collected data on the sentiments toward
unions of Southern nonunion textile laborers (n = 173). This example is
presented in the LISREL manual (Jöreskog & Sörbom, 1993, pp. 12–15,
example 3); included in the data files of the LISREL program; and was
utilized by Bollen (1989, pp. 82–83). The model consists of five observed
variables; the independent variables are the number of years worked in
the textile mill (actually log of years, denoted simply as years) and worker
age (age); the dependent variables are deference to managers (deference),
support for labor activism (support), and sentiment toward unions (sentiment). The original variance–covariance matrix, implied model (reproduced) variance–covariance matrix, residual matrix, and standardized
residual matrix are given in Table 7.1. The path diagram of the theoretical
proposed model is shown in Figure 7.1.
Path models adhere to certain common drawing conventions that are
utilized in SEM models (Figure 7.2). The observed variables are enclosed by
boxes or rectangles. Lines directed from one observed variable to another
observed variable denote direct effects, in other words, the direct influence
of one variable on another. For example, it is hypothesized that age has a
direct influence on support, meaning that the age of the worker may influence an increase (or decrease) in support. A curved, double-headed line
between two independent observed variables indicates covariance; that is,
they are correlated. In this example, age and years are specified to correlate.
The rationale for such relationships is that there are influences on both of
these independent variables outside of the path model. Because these influences are not studied in this path model, it is reasonable to expect that the
same unmeasured variables may influence both independent variables.

Y102005.indb 144

3/22/10 3:25:55 PM

145

Path Models

Table 7.1
Original, Reproduced, Residual, and Standardized Residual Covariance
Matrices for the Initial Union Sentiment Model
Original Matrix
Variable
Deference
Support
Sentiment
Years
Age

Deference

Support

Sentiment

Years

Age

14.610
−5.250
−8.057
−0.482
−18.857

11.017
11.087
0.677
17.861

31.971
1.559
28.250

1.021
7.139

215.662

Support

Sentiment

Years

Age

11.017
10.210
0.591
17.861

30.534
1.517
25.427

1.021
7.139

215.662

Support

Sentiment

Years

Age

0.000
0.877
0.086
0.000

1.437
0.042
2.823

0.000
0.000

0.000

Support

Sentiment

Years

Age

0.000
3.385
0.409
0.000

3.196
0.225
0.715

0.000
0.000

0.000

Reproduced Matrix
Variable
Deference
Deference
Support
Sentiment
Years
Age

14.610
−1.562
−5.045
−0.624
−18.857

Residual Matrix
Variable
Deference
Deference
Support
Sentiment
Years
Age

0.000
−3.688
−3.012
0.142
0.000

Standardized Residual Matrix
Variable
Deference
Deference
Support
Sentiment
Years
Age

0.000
–4.325
–3.991
0.581
0.000

Finally, each dependent variable has an error term, denoted by a circle
around the error term pointing toward the proper dependent variable.
Take deference, for example, some variance in deference scores will be predicted or explained by age and some variance will not. The unexplained
variance will become the error term, which indicates other possible influences on deference that are not contained in the specified path model.

Y102005.indb 145

3/22/10 3:25:55 PM

146

A Beginner’s Guide to Structural Equation Modeling

error1

deference

age

support

error2
sentiment

years

error3
Figure 7.1
Union sentiment model.
or

Latent variable

Observed variable

Unidirectional path

Disturbance or error in latent variable

Measurement error in observed variable

Correlation between variables
Recursive (nonreciprocal) relation between variables
Nonrecursive (reciprocal) relation between variables

Figure 7.2
Common path diagram symbols.

Y102005.indb 146

3/22/10 3:25:56 PM

147

Path Models

7.2 Model Specification
Model specification is necessary in examining multiple variable relationships in path models, just as in the case of multiple regression. Many different relationships among a set of variables can be hypothesized with
many different parameters being estimated. In a simple three-variable
model, for example, many possible path models can be postulated on the
basis of different hypothesized relationships among the three variables.
For example, in Figure 7.3a–c we see three different path models where
X1 influences X2. In Model (a), X1 influences X2, which in turn influences
Y. Here, X2 serves as a mediator between X1 and Y. In Model (b), an additional path is drawn from X1 to Y, such that X1 has both a direct and an
indirect effect upon Y. The direct effect is that X1 has a direct influence on
Y (no variables intervene between X1 and Y), whereas the indirect effect
is that X1 influences Y through X2, that is, X2 intervenes between X1 and
Y. In Model (c), X1 influences both X2 and Y; however, X2 and Y are not
related. If we were to switch X1 and X2 around, this would generate three
more plausible path models.
Other path models are also possible. For example, in Figure 7.4(a,b), X1
does not influence X2. In Model (a), X1 and X2 influence Y, but are uncorrelated. In Model (b), X1 and X2 influence Y and are correlated. How can
one determine which model is correct? This is known as model specification and shows the important role that theory and previous research plays
in justifying a hypothesized model. Path analysis does not provide a way
to specify the model, but rather estimates the effects among the variables
once the model has been specified a priori by the researcher on the basis of
theoretical considerations. For this reason, model specification is a critical
part of SEM modeling.
Path coefficients in path models are usually derived from the values of a
Pearson product moment correlation coefficient (r) and/or a standardized
partial regression coefficient (b) (Wolfle, 1977). For example, in the path
model of Figure 7.4b, the path coefficients (p) are depicted by arrows from
X1 to Y and X2 to Y, respectively, as:
b1 = pY1
b 2 = pY2
and the curved arrow between X1 and X2 is denoted as:
rX1,X2 = p12.
The variable relationships, once specified in standard score form, become
standardized partial regression coefficients. In multiple regression, a

Y102005.indb 147

3/22/10 3:25:56 PM

148

A Beginner’s Guide to Structural Equation Modeling

error1

error2

(a)

error1

error2

Y
(b)

error1

X2
X1
Y

error2
(c)
Figure 7.3
Possible three-variable models (X1 influences X2).

dependent variable is regressed in a single analysis on all of the independent variables. In path analysis, one or more multiple regression analyses
are performed depending on the variable relationships specified in the
path model. Path coefficients are therefore computed only on the basis
of the particular set of independent variables that lead to the dependent

Y102005.indb 148

3/22/10 3:25:56 PM

149

Path Models

error1

X2
(a)

error2

X2
(b)
Figure 7.4
Possible three-variable models (X1 does not influence X2).

variable under consideration. In the path model of Figure 7.4b, two standardized partial regression coefficients (path coefficients) are computed,
pY1 and pY2. The curved arrow represents the covariance or correlation
between the two independent variables p12 in predicting the dependent
variable.
For the union sentiment model, the model specification is as follows.
There are three structural equations in the model, one for each of the three
dependent variables, deference, support, and sentiment. In terms of variable names, the structural equations are as follows.
deference = age + error1
support = age + deference + error2
sentiment = years + support + deference + error3 .

Y102005.indb 149

3/22/10 3:25:57 PM

150

A Beginner’s Guide to Structural Equation Modeling

Substantive information from prior research suggested that those six
paths be included in the specified model; and that other possible paths,
for example from age to sentiment, not be included. This model includes
direct effects, for example from age to support, indirect effects, for example from age to support through deference, and correlated independent
variables, for example age and years. Obviously many possible path models could be specified for this set of observed variables.

7.3 Model Identification
Once a particular path model has been specified, the next concern is
whether the model is identified. In structural equation modeling, it is crucial that the researcher resolve the identification problem prior to the estimation of parameters. The general notion of identification was discussed in
Chapter 4. Here, we consider model identification in the context of path
models, and in particular, for our union sentiment example.
As described in Chapter 4, for the identification problem, we ask the following question: On the basis of the sample data contained in the sample
covariance matrix S and the theoretical model implied by the population
covariance matrix Σ, can a unique set of parameter estimates be found?
For the union sentiment model, for example, we would like to know if the
path between age and deference is identified; an example of one parameter to be estimated.
In the union sentiment model, some parameters are fixed and others
are free. An example of a fixed parameter is that there is no path or direct
relationship between age and sentiment. An example of a free parameter
is that there is a path or direct relationship between age and deference.
In determining identification, first consider the order condition. Here,
the number of free parameters to be estimated must be less than or equal
to the number of distinct values in the matrix S. In our path model we
specified the following:
6 path coefficients
3 equation error variances
1 correlation among the independent variables
2 independent variable variances
Thus, there are a total of 12 free parameters that we wish to estimate. The
number of distinct values in the matrix S is equal to:
[p (p + 1)]/2 = [5 (5 + 1)]/2 = 15,

Y102005.indb 150

3/22/10 3:25:57 PM

151

Path Models

where p is the number of observed variables in the matrix. Thus, the number of distinct values in the sample matrix S, 15 is indeed greater than the
number of free parameters, 12. However, this is only a necessary condition and does not guarantee that the model is identified. According to the
order condition, the model is also overidentified because there are more
values in S than parameters to be estimated.
Although the order condition is easy to assess, other sufficient conditions are not, for example, the rank condition. The sufficient conditions
require us to algebraically determine whether each parameter in the
model can be estimated from the covariance matrix S. According to the
LISREL computer program, which checks on identification, the union sentiment model is identified.

7.4 Model Estimation
Once the identification problem has been addressed, the next step is to estimate the parameters of the specified model. In this section, we consider the
following topics: decomposition of the correlation matrix, parameter estimation in general, and parameter estimation of the union sentiment model.
In path analysis, the traditional method of intuitively thinking about
estimation is to decompose the correlation matrix. This harkens back to
the early days of path analysis in the 1960s when sociologists like Arthur S.
Goldberger and Otis D. Duncan were rediscovering and further developing the procedure. The decomposition idea is that the original correlation
matrix can be completely reproduced if all of the effects are accounted for
in a specified path model. In other words, if all of the possible unidirectional (or recursive) paths are included in a path model, then the observed
correlation matrix can be completely reproduced from the obtained standardized estimates of the model.
For example, take the model in Figure 7.4b. Here there are two direct
effects, from X1 to Y and from X2 to Y. There are also indirect effects due
to the correlation between X1 and X2. In other words, X1 indirectly influences Y through X2, and also X2 indirectly affects Y through X1. The correlations among these three variables can be decomposed as follows:
r12 = p12
(CO)

(1)

r Y1 = pY1 + p12 pY2
(DE) (IE)

(2)

r Y2 = pY2 + p12 pY1,
(DE) (IE)

(3)

Y102005.indb 151

3/22/10 3:25:57 PM

152

A Beginner’s Guide to Structural Equation Modeling

where the r values are the actual observed correlations and the p values
are the path coefficients (standardized estimates). Thus, in equation (1),
the correlation between X1 and X2 is simply a function of the path, or
correlation relationship (CO), between X1 and X2. In equation (2), the
correlation between X1 and Y is a function of (a) the direct effect (DE)
of X1 on Y, and (b) the indirect effect (IE) of X1 on Y through X2 [the
product of the path or correlation between X1 and X2 (p12) and the path
or direct effect from X2 to Y (pY2)]. Equation (3) is similar to equation (2)
except that X1 and X2 are reversed; there is both a direct effect and an
indirect effect.
Let us illustrate how this works with an actual set of correlations. The
observed correlations are as follows: r12 = .224, r Y1 = .507, and r Y2 = .480. The
specified path model and correlation matrix were run in LISREL. The path
coefficients and the complete reproduction of the correlations are:
r12 = p12 = .224
(CO)

(4)

r Y1 = pY1 + p12 pY2 = .421 + (.224)(.386) = .507
(DE) (IE)

(5)

r Y2 = pY2 + p12 pY1 = .386 + (.224)(.421) = .480.
(DE) (IE)

(6)

Here, the original correlations are completely reproduced by the model
because all of the effects are accounted for, direct, indirect, and correlated.
If a path were left out of the model, for example p12, then the correlations
would not be completely reproduced. Thus, the correlation decomposition
approach is a nice conceptual way of thinking about the estimation process in path analysis. For further details on the correlation decomposition
approach, we highly recommend reading Duncan (1975).
In chapter 4, we presented the problem of estimation in general.
Parameters can be estimated by different estimation procedures, such
as maximum likelihood (ML), generalized least squares (GLS), and
unweighted least squares (ULS), which are all unstandardized types of
estimates, as well as standardized estimates (the path coefficients previously described in this chapter were standardized estimates). In addition
to different methods of estimation of the parameter estimates, full versus
limited information estimation functions are invoked based on the software chosen for the analysis. Full information estimation computes all of the
parameters simultaneously, whereas limited information estimation computes parameters for each equation separately. The parameters estimated
in structural equation modeling software (LISREL) use full information
estimation and therefore differ from parameter estimates computed in

Y102005.indb 152

3/22/10 3:25:57 PM

153

Path Models

SPSS or SAS, where each equation in the path model is estimated separately (limited information estimation). In limited information estimation,
the parameter estimates are determined uniquely in each separate equation to meet the least squares criterion of minimized residuals.
In the union sentiment example we see the estimation process at work. In
order to utilize the model modification procedures discussed in section 7.6,
we have slightly changed the model specification in Figure 7.1. We remove
the path from deference to support and call this the initial model. We evaluate this initial model, and hope, through the model modification process,
we will obtain the model as originally specified in Figure 7.1. The intentionally misspecified model was run using LISREL (Note: The LISREL program
for the correctly specified model is given at the end of the chapter).
The maximum likelihood estimates for the initial model are shown in the
first column of Table 7.2. All of the parameter estimates are significantly
Table 7.2
Maximum Likelihood Estimatesa and Selected Fit Indices for the Initial
and Final Union Sentiment Models
Paths

Final Model

Age → deference
Age → support
Deference → support
Years → sentiment
Deference → sentiment
Support → sentiment

−.09
0.08
—
0.86
−.22
0.85

−.09
0.06
−.28
0.86
−.22
0.85

Equation error variances
Deference
Support
Sentiment

12.96
9.54
19.45

12.96
8.49
19.45

Independent variables
Variance (age)
Variance (years)
Covariance (age, years)

215.66
1.02
7.14

Selected fit indices
c2
df
p value
RMSEA
SRMR
GFI
a

Y102005.indb 153

Initial Model

19.96
4
.00
.15
.087
.96

1.25
3
.74
.00
.015
1.00

All estimates significantly different from zero (p < .05).

3/22/10 3:25:57 PM

154

A Beginner’s Guide to Structural Equation Modeling

different from zero, p < .05 (the fit of the model is discussed next in section 7.5). Age has a direct effect on both deference and support; deference
has a direct effect on sentiment; years has a direct effect on sentiment;
and support has a direct effect on sentiment. Numerous indirect effects
are also part of the path model, such as the indirect effect of age on sentiment through support. Age and years also have a significant covariance,
indicating that one or more common unmeasured variables influence both
age and years.

7.5 Model Testing
An important result of any path analysis is the fit of the specified
model. If the fit of the path model is good, then the specified model has
been supported by the sample data. If the fit of the path model is not so
good, then the specified model has not been supported by the sample
data, and the researcher typically attempts to modify the path model to
achieve a better fit (as described in section 7.6). As discussed in chapter 5, LISREL provides modification indices and expected parameter
changes values to guide modifying a model to obtain better model-fit
criteria.*
For purposes of the union sentiment example, we include a few modelfit indices at the bottom of Table 7.2. For the initial path model, the c2
statistic, technically a measure of badness of fit, is equal to 19.96, with four
degrees of freedom, and p < .01. As the p value is very small and the c2
value is nowhere near the number of degrees of freedom, then according
to this measure of fit, the initial path model is poorly specified. The rootmean-square error of approximation (RMSEA) is equal to .15, somewhat
below the acceptable level for this measure of fit (RMSEA < .08 or .05). The
standardized root-mean-square residual (SRMR) is .087, also below the
usual acceptable level of fit (SRMR < .08 or .05). Finally, the goodness-of-fit
index (GFI) is .96 for the initial model, which is an acceptable level for this
measure of fit (GFI > .95). Across this particular set of model-fit indices,
the conclusion is that the data to model fit is approaching a reasonable
level, but that some model modifications might allow us to achieve a better model fit between the sample variance–covariance matrix S and the
implied model (reproduced) variance–covariance matrix Σ. Model modification is considered in the next section.

Another traditional non-SEM path model-fit index is described in the Chapter Footnote.

Y102005.indb 154

3/22/10 3:25:58 PM

Path Models

155

7.6 Model Modification
The final step in structural equation modeling is model modification. In
other words, if the fit of the model is less than satisfactory, then the researcher
typically performs a specification search to seek a better fitting model. As
described in chapters 4 and 5, several different procedures can be used to
assist in this search. One may eliminate parameters that are not significantly different from zero and/or include additional parameters to arrive at
a modified model. For the elimination of parameters, the most commonly
used procedure in LISREL is to compare the t statistic for each parameter to
a tabled t value (e.g., t > 1.96) to determine statistical significance.
For the inclusion of additional parameters, the most commonly used
techniques in LISREL are (a) the modification index (MI) (the expected
value that c2 would decrease if such a parameter were to be included;
large values indicate potentially useful parameters), and (b) the expected
parameter change statistic (EPC) (the approximate value of the new
parameter if added to the model).
In addition, an examination of the residual matrix, or the more useful standardized residual matrix, often gives clues as to which original covariance or
correlations are not well accounted for by the model. Recall that the residual
matrix is the difference between the observed variance-covariance S and the
model implied (reproduced) variance-covariance matrix Σ. Large residuals
indicate values not well accounted for by the model. Standardized residuals are like z scores in that large values (greater than 1.96 or 2.58) indicate
that a particular relationship is not well accounted for by the path model
(Table A.1).
For the initial union sentiment example, the original, model implied
(reproduced), residual, and standardized residual covariance matrices are
given in Table 7.1. Here we see that the largest standardized residual is
between deference and support (−4.325). The t statistics do not suggest
the elimination of any existing parameters from the initial path model
because every parameter is statistically different from zero. With regard
to the possible inclusion of new parameters, the largest modification index
is for the path from deference to support (MI = 18.9). For that potential
path, the estimated value, or expected parameter change (EPC), is −0.28.
Taken together, these statistics indicate that there is something misspecified between deference and support that is not captured by the initial
model. Specifically, adding a path is recommended from deference to support. This is precisely the path from the originally specified path model
that we intentionally eliminated from the initial path model. Thus, the
specification search was successful in obtaining the original model. The
ML estimates and selected fit indices for the final model, where this path
is now included, are shown in the second column of Table 7.2. All of the

Y102005.indb 155

3/22/10 3:25:58 PM

156

A Beginner’s Guide to Structural Equation Modeling

parameters included are significantly different from zero (p < .05), all of the
fit indices now indicate an acceptable level of fit, and no additional modification indices are indicated for any further recommended changes. Thus,
we deem this as the final path model for the union sentiment example.

7.7 Summary
This chapter presented a detailed discussion of path models. We began by
presenting the union sentiment path model and then followed it throughout the chapter. We moved on to model specification, first with several
possible three-variable models, and then with the union sentiment model.
The next step was to consider model identification of the union sentiment
model for both the order and rank conditions. Next, we discussed estimation. Here, we introduced the notion of correlation decomposition with
a three-variable model, and the difference between full versus limited
estimation functions, and then considered the full information estimation
results for the union sentiment model. Model testing of the misspecified
union sentiment model was the next step, where the fit of the model was
deemed not acceptable. The misspecified model (altered initial model) was
then modified through the addition of one path, thereby arriving at a final,
best-fitting theoretical model, which was the same as our initial model.
We learned that path models permit theoretically meaningful relationships among variables that cannot be specified in a single additive regression model. However, the issue of measurement error in observed variables
is not treated in either regression or path models (Wolfle, 1979). The next
chapter helps us to understand how measurement error is addressed in
structural equation modeling via factor models.

Appendix: LISREL–SIMPLIS Path
Model Program (Figure 7.1)
Union Sentiment of Textile Workers
Observed Variables: Deference Support Sentiment Years Age
Covariance matrix:
14.610
−5.250 11.017
−8.057 11.087 31.971
−0.482 0.677 1.559 1.021
−18.857 17.861 28.250 7.139 215.662

Y102005.indb 156

3/22/10 3:25:58 PM

157

Path Models

Sample Size: 173
Relationships
Deference = Age
Support = Age Deference
Sentiment = Years Deference Support
Print Residuals
Options: ND = 3
Path Diagram
End of Problem
Exercise
1. Analyze the following achievement path model (Figure 7.5)
using the LISREL software program. The path model indicates
that income and ability predict aspire, and income, ability, and
aspire predict achieve.
Sample size = 100
Observed variables: quantitative achievement (Ach), family
income (Inc), quantitative ability (Abl), educational aspiration (Asp)
Variance–covariance matrix:

Ach
Inc
Abl
Asp

Ach

Inc

Abl

Asp

25.500
20.500
22.480
16.275

38.100
24.200
13.600

42.750
13.500

17.000

Equations:
Asp = Inc Abl
Ach = Inc Abl Asp

Income

error1

Aspire

error2

Achieve

Ability
Figure 7.5
Achievement path model.

Y102005.indb 157

3/22/10 3:25:59 PM

158

A Beginner’s Guide to Structural Equation Modeling

Chapter Footnote
Another Traditional Non-SEM Path Model-Fit Index
The relationship between the original and reproduced correlation matrices is essential for testing the significance of the path model (Specht, 1975).
The relationship between the two matrices is tested by calculating a chisquare statistic. A significant c2 value for a specified level of significance
(a = .05) indicates that the path model does not fit the data. If c2 = 0, then
the original and reproduced correlations in the matrices are identical; in
other words, the correlations are perfectly reproduced by the path model.
Also, if the residuals, for example Pe1 and Pe2, are uncorrelated in a path
model, then the sum of squared residual path coefficients will equal the
chi-square value. A non-significant chi-square value therefore indicates a
good path model to data fit in SEM. Another traditional non-SEM path
model-fit index, Q, has been reported in the research literature and therefore presented here using a LISREL–SIMPLIS program example with heuristic data.
LISREL–SIMPLIS program
Path analysis of Y
Observed variables Y X1 X2 X3
Sample size 100
Correlation Matrix
1.000
.507 1.000
.480 .224 1.000
.275 .062 .577 1.000
Equation:
Y = X1 X2 X3
X3 = X1 X2
End of Problem

The theoretical path model in Figure 7.6 indicates that two variables,
X1 and X2 predict X3; X1, X2, and X3 predict Y; and X1 and X2 are correlated. This original path model is a saturated model because all paths
are included, thus c2 = 0, df = 0, and p = 0. The original path model,
however, has two R-squared values for each regression equation: R 2X3.
2
X1,X2 = .34 and R Y.X3,X1,X2 = .40. The path model diagram only shows the
1 − R 2X3.X1,X2 = .66 and 1 − R 2Y.X3,X1,X2 = .60 values. Computer output indicated that the path from X1 to X3 was non-significant ( p 31 = −.071) and
the path from X3 to Y was non-significant (p 3Y = .040). For theoretical
reason, we only dropped path p 31 resulting in the modified path model
in Figure 7.7.

Y102005.indb 158

3/22/10 3:25:59 PM

159

Path Models

X1
0.423

–0.071
0.224

X3
0.593
X2

0.040

0.362
0.60

0.66

Figure 7.6
Original path model.

The modified path model reported a non-significant c2 = .71, df = 1, and
p = .40 which indicates that the data fits the path model, although the path
coefficient from X3 to Y is still non-significant, but kept in the model for
theoretical reasons.
The other traditional non-SEM path model-fit indices can be accomplished by computing the generalized squared multiple correlation
(Pedhazur, 1982) as follows:
Rm2 = 1 – (1 – R12) (1 − R22 )…..(1 − Rp2 ).
The R-squared values are the squared multiple correlation coefficients
from each of the separate regression analyses in the path model. In the
original path model, the two regression analyses yielded R-squared values
of .34 and .40, respectively. The path model-fit R 2m would be computed as:
Rm2 = 1 – (1 − .34)(1 − .40) = .604

X1
0.42
0.224

X3
0.58
X2

0.04

0.36
0.67

0.60

Figure 7.7
Modified path model.

Y102005.indb 159

3/22/10 3:26:00 PM

160

A Beginner’s Guide to Structural Equation Modeling

An analogous statistic to Rm2 , M, and a large sample measure of model
fit, Q, are also presented in Pedhazur (1982). Q is oftentimes recommended
because chi-square is affected by sample size. Q varies between zero and one
and is not a function of sample size. The formula for Q is:
Q = (1− Rm2 )/(1 – M)
M is calculated in the same manner as Rm2 , but with a non-significant path
deleted. In our example, we dropped the path from X1 to X3 because it
yielded a non-significant path coefficient and therefore M calculated from
the modified path model would have a different value from Rm2 in the original path model (M values range between zero and Rm2 ).
In our example, the path from X1 to X3 in the program was dropped by
changing the first LISREL–SIMPLIS equation command to read:
Equation: X3 = X2
The M value is computed as:
M = 1 – (1 − .33)(1 − .40) = .598.
Q is now computed as:
Q = [(1 − .604)/( 1 − .598)] = [.396/.402] = .98.
Remember, the closer the value of Q to 1.0, the better the model fit. Q can
be tested for significance using W, which is computed as:
W = − (N – d) loge Q,
where N = sample size, d = number of path coefficients hypothesized to be
zero, loge = natural logarithm (ln). For our example,
W = − (100 − 1) loge (.98) = 2.00.
Since W approximates the c2 distribution with degrees of freedom = d,
the tabled critical chi-square value for d = 1, a = 05, is 3.841 (Table A.4).
W is less than the tabled critical value, therefore nonsignificant, suggesting a good path model fit to the data. The W value fell between p = .20
(c2 = 1.642) and p = .10 (c2 = 2.706) in Table A.4.
Prior to SEM, Q and the W path model-fit index were reported to test
whether a path model significantly reproduced the correlation matrix. The
R2m value was reported to indicate the amount of variation in Y predicted
by the direct and indirect effects of the independent variables. Individual

Y102005.indb 160

3/22/10 3:26:00 PM

Path Models

161

tests of path coefficients were also computed and reported by dividing
the path coefficient by its standard error. We used the path coefficient,
standard error, and associated t-value provided in the computer output to
determine if a path coefficient was nonsignificant, thus dropping it from
the path model.

References
Bollen, K. A. (1989). Structural equations with latent variables. New York: Wiley.
Duncan, O. D. (1975). Introduction to structural equation models. New York: Academic.
Jöreskog, K. G., & Sörbom, D. (1993). LISREL 8: Structural equation modeling with the
SIMPLIS command language. Chicago, IL: Scientific Software International.
McDonald, J. A., & Clelland, D. A. (1984). Textile workers and union sentiment.
Social Forces, 63, 502–521.
Pearl, J. (2009). Causality: Models, reasoning, and inference (2nd ed.). Cambridge
University Press: London.
Pedhazur, E. J. (1982). Multiple regression in behavioral research: Explanation and prediction (2nd ed.). New York: Holt, Rinehart & Winston.
Specht, D. A. (1975). On the evaluation of causal models. Social Science Research,
4, 113–133.
Wolfle, L. M. (1977). An introduction to path analysis. Multiple Linear Regression
Viewpoints, 8, 36–61.
Wolfle, L. M. (1979). Unmeasured variables in path analysis. Multiple Linear
Regression Viewpoints, 9, 20–56.
Wright, S. (1921). Correlation and causation. Journal of Agricultural Research, 20,
557–585.
Wright, S. (1934). The method of path coefficients. Annals of Mathematical Statistics,
5, 161–215.
Wright, S. (1960). Path coefficients and path regression: Alternative or complementary concepts? Biometrics, 16, 189–202.

Y102005.indb 161

3/22/10 3:26:00 PM

8
Confirmatory Factor Models

Key Concepts
Confirmatory factor analysis versus exploratory factor analysis
Latent variables (factors) and observed variables
Factor loadings and measurement errors
Correlated factors and correlated measurement errors
In chapter 7 we examined path models as the logical extension of multiple
regression models (chapter 6) to show more meaningful theoretical relationships among our observed variables. Thus, the two previous chapters
dealt exclusively with models involving observed variables. In this chapter we begin developing models involving factors or latent variables and
continue latent variable modeling throughout the remainder of the book.
As we see in this chapter, a major limitation of models involving only
observed variables is that measurement error is not taken into account.
The use of observed variables in statistics assumes that all of the measured variables are perfectly valid and reliable, which is unlikely in many
applications. For example, father’s educational level is not a perfect measure of a socioeconomic status factor and amount of exercise per week is
not a perfect measure of a fitness factor.
The validity and reliability issues in measurement have traditionally
been handled by first examining the validity and reliability of scores on
instruments used in a particular context. Given an acceptable level of score
validity and reliability, the scores are then used in a statistical analysis.
However, the traditional statistical analysis of these scores—for example,
in multiple regression and path analysis—does not adjust for measurement error. The impact of measurement error has been investigated and
found to have serious consequences—for example, biased parameter estimates (Cochran, 1968; Fuller, 1987). Structural equation modeling software that accounts for the measurement error of variables was therefore
developed—that is, factor analysis—which creates latent variables used in
structural equation modeling.
163

Y102005.indb 163

3/22/10 3:26:00 PM

164

A Beginner’s Guide to Structural Equation Modeling

Factor analysis attempts to determine which sets of observed variables
share common variance–covariance characteristics that define theoretical constructs or factors (latent variables). Factor analysis presumes that
some factors that are smaller in number than the number of observed
variables are responsible for the shared variance–covariance among the
observed variables. In practice, one collects data on observed variables
and uses factor-analytic techniques to either confirm that a particular
subset of observed variables define each construct or factor, or explore
which observed variables relate to factors. In exploratory factor model
approaches, we seek to find a model that fits the data, so we specify different alternative models, hoping to ultimately find a model that fits the data
and has theoretical support. This is the primary rationale for exploratory
factor analysis (EFA). In confirmatory factor model approaches, we seek to
statistically test the significance of a hypothesized factor model—that is,
whether the sample data confirm that model. Additional samples of data
that fit the model further confirm the validity of the hypothesized model.
This is the primary rationale for confirmatory factor analysis (CFA).
In CFA, the researcher specifies a certain number of factors, which factors
are correlated, and which observed variables measure each factor. In EFA,
the researcher explores how many factors there are, whether the factors are
correlated, and which observed variables appear to best measure each factor. In CFA, the researcher has an a priori specified theoretical model; in
EFA, the researcher does not have such a model. In this chapter we only
concern ourselves with confirmatory factor models because the focus of the
book is on testing theoretical models; exploratory factor analysis is covered
in depth elsewhere (Comrey & Lee, 1992; Gorsuch, 1983; and Costello &
Osborne, 2005). This chapter begins with a classic example of a confirmatory
factor model and then proceeds with sections on model specification, model
identification, model estimation, model testing, and model modification.

8.1 An Example
We use a classic confirmatory factor model that will be followed throughout the chapter. Holzinger and Swineford (1939) collected data on 26 psychological tests from seventh- and eighth-grade children in a suburban
school district of Chicago. Over the years, different subsamples of the
children and different subsets of the variables of this dataset have been
analyzed and presented in various multivariate statistics textbooks—for
example, Harmon (1976) and Gorsuch (1983)—and SEM software program
guides—for example, Jöreskog and Sörbom (1993; example 5, pp. 23–28).
The raw data analyzed here are on the first six psychological variables
for all 301 subjects (see chapter 5); the resulting sample covariance matrix

Y102005.indb 164

3/22/10 3:26:00 PM

165

Confirmatory Factor Models

spatial

verbal

visperc

err_v

cubes

err_c

lozenges

err_l

paragrap

err_p

sentence

err_s

wordmean

err_w

Figure 8.1
Confirmatory factor model. (From Holzinger, K. J., & Swineford, F. A. [1939]. A study in factor
analysis: The stability of a bi-factor solution. [Supplementary Educational Monographs, No. 48].
Chicago, IL: University of Chicago, Department of Education.)

S is given in the Appendix. The confirmatory factor model consists of
the following six observed variables: Visual Perception, Cubes, Lozenges,
Paragraph Comprehension, Sentence Completion, and Word Meaning. The
first three measures were hypothesized to measure a spatial ability factor and the second three measures to measure a verbal ability factor.
The path diagram of the theoretical proposed model is shown in
Figure 8.1. The drawing conventions utilized in Figure 8.1 were described
in chapter 7. The observed variables are enclosed by boxes or rectangles,
and the factors (latent variables) are enclosed by circles or ellipses—that is,
spatial and verbal. Conceptually, a factor represents the common variation
among a set of observed variables. Thus, for example, the spatial ability
factor represents the common variation among the Visual Perception, Cubes,
and Lozenges tasks. Lines directed from a factor to a particular observed
variable denote the relationship between that factor and that measure.
These relationships are interpreted as factor loadings with the square of
the factor loading called the commonality estimate of the variable.
The measurement errors are enclosed by smaller ellipses and indicate
that some portion of each observed variable is measuring something
other than the hypothesized factor. Conceptually, a measurement error

Y102005.indb 165

3/22/10 3:26:01 PM

166

A Beginner’s Guide to Structural Equation Modeling

represents the unique variation for a particular observed variable beyond
the variation due to the relevant factor. For example, the Cubes task is
largely a measure of spatial ability, but may also be assessing other characteristics such as a different common factor or unreliability. To assess
measurement error, the variance of each measurement error is estimated
(known as measurement error variance).
A curved, double-headed line between two factors indicates that they
have shared variance or are correlated. In this example, spatial and verbal
ability are specified to covary or correlate. The rationale for this particular
factor correlation is that spatial ability and verbal ability are related to a
more general ability factor and thus should be theoretically related.
A curved, double-headed line between two measurement error variances indicates that they also have shared variance or are correlated.
Although not shown in this example, two measurement error variances
could be correlated if they shared something in common such as (a) common method variance where the method of measurement is the same, such
as the same scale of measurement, or they are both part of the same global
instrument, or (b) the same measure is being used at different points in
time, that is, the Cubes task is measured at Time 1 and again at Time 2.

8.2 Model Specification
Model specification is a necessary first step in analyzing a confirmatory
factor model, just as it was for multiple regression and path models. Many
different relationships among a set of variables can be postulated with
many different parameters being estimated. Thus, many different factor
models can be postulated on the basis of different hypothesized relationships between the observed variables and the factors.
In our example, there are six observed variables with two different
latent variables (factors) being hypothesized. Given this, many different confirmatory factor models are possible. First, each observed
variable can load on either one or both factors. Thus, there could be
anywhere from 6 to 12 total factor loadings. Second, the two factors
may or may not be correlated. Third, there may or may not be correlations or covariance terms among the measurement error variances.
Thus, there could be anywhere from 0 to 15 total correlated measurement error variances.
From the model in Figure 8.1, each observed variable is hypothesized to
measure only a single factor—that is, three observed variables per factor
with six factor loadings; the factors are believed to be correlated (a single factor correlation); and the measurement error variances are not related (zero

Y102005.indb 166

3/22/10 3:26:01 PM

Confirmatory Factor Models

167

correlated measurement errors. Obviously, we could have hypothesized a
single factor with six observed variables or six factors each with a single
observed variable. When all of this is taken into account, many different
confirmatory factor models are possible with these six observed variables.
How does the researcher determine which factor model is correct? We
already know that model specification is important in this process and indicates the important role that theory and prior research play in justifying
a specified model. Confirmatory factor analysis does not tell us how to
specify the model, but rather estimates the parameters of the model once
the model has been specified a priori by the researcher on the basis of
theoretical and research based knowledge. Once again, model specification is the hardest part of structural equation modeling.
For our confirmatory factor model, the model specification is diagrammed in Figure 8.1 and contains six measurement equations in the
model, one for each of the six observed variables. In terms of the variable
names from Figure 8.1, the measurement equations are as follows:
visperc = function of spatial + err_v
cubes = function of spatial + err_c
lozenges = function of spatial + err_l
paragrap = function of verbal + err_p
sentence = function of verbal + err_s
wordmean = function of verbal + err_w
Substantive theory and prior research suggest that these particular factor
loadings should be included in the specified model (the functions being the
factor loadings), and that other possible factor loadings—for example, visperc loading on verbal, should not be included in the factor model. Our factor model includes six factor loadings and six measurement error variances,
one for each observed variable, and one correlation between the factors spatial ability and verbal ability with zero correlated measurement errors.

8.3 Model Identification
Once a confirmatory factor model has been specified, the next step is
to determine whether the model is identified. As stated in chapter 4, it
is crucial that the researcher solve the identification problem prior to the

Y102005.indb 167

3/22/10 3:26:01 PM

168

A Beginner’s Guide to Structural Equation Modeling

estimation of parameters. We first need to revisit model identification in
the context of confirmatory factor models and then specifically for our
confirmatory factor model example.
In model identification (see chapter 4), we ask the following question:
On the basis of the sample data contained in the sample variance–
covariance matrix S, and the theoretical model implied by the population
variance–covariance matrix Σ, can a unique set of parameter estimates
be found? For our confirmatory factor model, we would like to know if
the factor loading of Visual Perception on Spatial Ability, Cubes on Spatial
Ability, Lozenges on Spatial Ability, Paragraph Comprehension on Verbal Ability,
Sentence Completion on Verbal Ability, and Word Meaning on Verbal Ability
are identified (can be estimated). In our confirmatory factor model, some
parameters are fixed and others are free. An example of a fixed parameter
is that Cubes is not allowed to load on Verbal Ability. An example of a free
parameter is that Cubes is allowed to load on Spatial Ability.
In determining identification, we first assess the order condition. The
number of free parameters to be estimated must be less than or equal to
the number of distinct values in the matrix S. A count of the free parameters is as follows:
6 factor loadings
6 measurement error variances
0 measurement error covariance terms or correlations
1 correlation among the latent variables
Thus, there are a total of 13 free parameters that we wish to estimate. The
number of distinct values in the matrix S is equal to
p (p + 1)/2 = 6 (6 + 1)/2 = 21,
where p is the number of observed variables in the sample variance–
covariance matrix. The number of values in S, 21, is greater than the
number of free parameters, 13, with the difference being the degrees of
freedom for the specified model, df = 21 − 13 = 8. However, this is only
a necessary condition and does not guarantee that the model is identified. According to the order condition, this model is over-identified because
there are more values in S than parameters to be estimated—that is, our
degrees of freedom is positive not zero (just-identified) or negative (underidentified).
Although the order condition is easy to assess, other sufficient conditions are not, for example, the rank condition. The sufficient conditions
require us to algebraically determine whether each parameter in the
model can be estimated from the covariance matrix S. According to

Y102005.indb 168

3/22/10 3:26:02 PM

Confirmatory Factor Models

169

the LISREL computer program, which checks on identification through
the rank test and information matrix, the confirmatory factor model is
identified.

8.4 Model Estimation
After the identification problem has been addressed, the next step is to
estimate the parameters of the specified factor model. In this section we
consider the following topics: decomposition of the correlation (or variance–
covariance) matrix, parameter estimation in general, and parameter estimation for the confirmatory factor model example.
In factor analysis the traditional method of intuitively thinking about
estimation is to decompose the correlation (or variance–covariance)
matrix. The decomposition notion is that the original correlation (or variance–covariance) matrix can be completely reproduced if all of the relations among the observed variables are accounted for by the factors in
a properly specified factor model. If the model is not properly specified,
then the original correlation (or variance–covariance) matrix will not be
completely reproduced. This would occur if (a) the number of factors was
not correct, (b) the wrong factor loadings were specified, (c) the factor correlations were not correctly specified, and/or (d) the measurement error
variances were not specified correctly.
In chapter 4, under model estimation, we considered the statistical
aspects of estimation. We learned, for example, that parameters can be
estimated by different estimation procedures, such as maximum likelihood (ML), generalized least squares (GLS), and unweighted least squares
(ULS), and reported as unstandardized estimates or standardized estimates. We analyzed our confirmatory factor model using maximum likelihood estimation with a standardized solution to report our statistical
estimates of the free parameters.
To better understand model modification in section 8.6, we have slightly
changed the confirmatory factor model specified in Figure 8.1. We forced
the observed variable Lozenges to have a factor loading on the latent variable Verbal Ability instead of on the latent variable Spatial Ability. This intentionally misspecified model is shown in Figure 8.2. We therefore use the
confirmatory factor model in Figure 8.2 as our initial model and through
the model modification process in section 8.6 hope to discover the bestfitting model to be the confirmatory factor model originally specified in
Figure 8.1.
The misspecified model (Figure 8.2) was run using LISREL (computer
program in chapter Appendix). The sample variance-covariance matrix S

Y102005.indb 169

3/22/10 3:26:02 PM

170

A Beginner’s Guide to Structural Equation Modeling

spatial

verbal

visperc

err_v

cubes

err_c

lozenges

err_l

paragrap

err_p

sentence

err_s

wordmean

err_w

Figure 8.2
Misspecified confirmatory factor model. (From Holzinger, K. J., & Swineford, F. A. [1939].
A study in factor analysis: The stability of a bi-factor solution. [Supplementary Educational
Monographs, No. 48]. Chicago, IL: University of Chicago, Department of Education.)

is given at the top of Table 8.1 along with the model implied (reproduced)
matrix, residual matrix, and standardized residual matrix for the misspecified model in Figure 8.2.
The first column in Table 8.2 contains the standardized estimates for
the misspecified model (Figure 8.2), and the second column contains the
standardized estimates for the original model (Figure 8.1). The parameter
estimates are found to be significantly different from zero (p < .05). The fit
of the model is discussed in section 8.5. Of greatest importance is that all
of the factor loadings are statistically significantly different from zero and
have the expected sign, that is, positive factor loadings.

8.5 Model Testing
An important part of the estimation process in analyzing confirmatory
factor models is to fit the sample variance–covariance data to the specified
model. If the fit of the model is good, then the specified model is supported

Y102005.indb 170

3/22/10 3:26:02 PM

171

Confirmatory Factor Models

Table 8.1
Original, Reproduced, Residual, and Standardized Residual Covariance
Matrices for the Misspecified Holzinger–Swineford Model
Original Matrix:
Variable

Visperc

Cubes

Visperc
Cubes
Lozenges
Parcomp
Sencomp
Wordmean

49.064
9.810
27.928
9.117
10.610
19.166

Lozenges

Parcomp

Sencomp

Wordmean

22.182
14.482
2.515
3.389
6.954

81.863
5.013
3.605
13.716

12.196
13.217
18.868

26.645
28.502

58.817

Lozenges

Parcomp

Sencomp

Wordmean

81.863
5.266
7.747
11.376

12.196
13.061
19.180

26.645
28.218

58.817

Lozenges

Parcomp

Sencomp

Wordmean

0.000
−0.253
−4.142
2.339

0.000
0.155
−0.312

0.000
0.283

0.000

Standardized Residual Matrix:
Variable
Visperc
Cubes
Lozenges

Parcomp

Sencomp

Wordmean

Visperc
Cubes
Lozenges
Parcomp
Sencomp
Wordmean

0.000
2.310
−2.861

0.000
1.696

0.000

Reproduced Matrix:
Variable
Visperc
Visperc
Cubes
Lozenges
Parcomp
Sencomp
Wordmean

49.064
9.810
5.098
8.595
12.646
18.570

Residual Matrix:
Variable
Visperc
Visperc
Cubes
Lozenges
Parcomp
Sencomp
Wordmean

0.000
0.000
22.830
0.522
−2.036
0.596

0.000
0.000
7.093
1.002
−2.587
0.484

Cubes
22.182
1.646
2.775
4.083
5.996
Cubes
0.000
12.836
−0.260
−0.694
0.958

0.000
5.455
−0.668
−1.182
1.046

0.000
−0.336
−3.647
1.321

by the sample data. If the fit of the model is not so good, then the specified
model is not supported by the sample data, and the researcher typically
has to modify the model to achieve a better fit (see section 8.6). As previously discussed in chapter 5, there is a wide variety of model-fit indices
available to the SEM researcher.

Y102005.indb 171

3/22/10 3:26:02 PM

172

A Beginner’s Guide to Structural Equation Modeling

Table 8.2
Standardized Estimates and Selected Fit Indices for the
Misspecified and Original Holzinger–Swineford Models
Misspecified Model

Original Model

Factor loadings:
Visual Perception
Cubes
Lozenges
Paragraph Comprehension
Sentence Completion
Word Meaning

.79
.38
.20
.85
.85
.84

.78
.43
.57
.85
.85
.84

Measurement error variances:
Visual Perception
Cubes
Lozenges
Paragraph Comprehension
Sentence Completion
Word Meaning

.38
.86
.96
.27
.28
.30

.39
.81
.68
.27
.27
.30

Correlation of independent variables:
(Spatial, Verbal)
.52

.46

Selected fit Indices:
c2
Df
p value
RMSEA
GFI

80.926
8
.001
.174
.918

24.407
8
.002
.083
.974

For our confirmatory factor model example, we report a few fit indices at
the bottom of Table 8.2. For the misspecified model, the c2 statistic (technically a measure of badness of fit) is equal to 80.926, with eight degrees of
freedom, and p < .001. The chi-square statistic is significant, so the specified confirmatory factor model is not supported by the sample variance–
covariance data. Another interpretation is that because the c2 value is not
close to the number of degrees of freedom, the fit of the initial model is poor.
Recall that the noncentrality parameter (NCP) is calculated as c2 – df, has
an expected value of 0 (NCP = 0; perfect fit), and is used in computing several of the model-fit indices. A third criterion is that the root-mean-square
error of approximation (RMSEA) is equal to .174, higher than the acceptable
level of model fit (RMSEA < .08 or .05). Finally, the goodness-of-fit index
(GFI) is .918 for the misspecified model, which is below the acceptable

Y102005.indb 172

3/22/10 3:26:03 PM

Confirmatory Factor Models

173

range of model fit (GFI > .95). Across this particular set of model-fit indices,
the conclusion is that the model fit is reasonable, although still not acceptable, but that some model modification might allow us to achieve a better
sample data (variance–covariance matrix) to confirmatory factor model fit.
Determining what change(s) to make to our confirmatory factor model to
achieve a better fitting model is considered in the next section.

8.6 Model Modification
A final step in structural equation modeling is to consider changes to a
specified model that has poor model-fit indices—that is, model modification. This typically occurs when a researcher discovers that the fit of the
specified model is less than satisfactory. The researcher typically performs
a specification search to find a better fitting model. As discussed in chapter 4, several different procedures can be used to assist in this specification
search. One may eliminate parameters that are not significantly different
from zero and/or include additional parameters to arrive at a modified
model. For the elimination of parameters, the most commonly used procedure in LISREL is to compare the t statistic for each parameter to a tabled t
value—for example t = 1.96, at a = .05, two-tailed test; or t = 2.58 at a = .01,
two-tailed test (see Table A.2), to determine statistical significance.
For the inclusion of additional parameters, the most commonly used
techniques in LISREL are (a) the modification index (MI—the expected
value that c2 would decrease if such a parameter were to be included in
the model; large values indicate potentially useful parameters), and (b) the
expected parameter change statistic (EPC—the approximate value of the
new parameter).
In addition, an examination of the residual matrix, or the more useful
standardized residual matrix, often gives clues as to which original covariance terms or correlations are not well accounted for by the model. The
residual matrix is the difference between the observed covariance or correlation matrix S and the model implied (reproduced) covariance or correlation matrix Σ. Large residuals indicate values not well accounted for by
the model. Standardized residuals are like z scores such that large values
(values greater than 1.96 or 2.58) indicate that a particular relationship is
not well accounted for by the model.
For the misspecified confirmatory factor model in Figure 8.2, the original, model-implied (reproduced), residual, and standardized residual
covariance matrices are given in Table 8.1. Here, we see that the two largest residuals are for the Lozenges observed variable (22.830 and 12.836) and
the standardized residuals (7.093 and 5.455) are greater than t = 1.96 or
2.58. The results also indicate that the Lozenges variable should load on the

Y102005.indb 173

3/22/10 3:26:03 PM

174

A Beginner’s Guide to Structural Equation Modeling

Spatial Ability factor to reduce error (MI = 60.11) with an expected parameter change (EPC) of 6.30.
The large residuals for Lozenges, the statistically significant standardized residuals, the modification index, and the expected change value all
indicated that there was something wrong with the Lozenges observed
variable that is not captured by the misspecified model. Specifically, the
factor loading for Lozenges should be on the Spatial Ability factor rather
than the Verbal Ability factor. This is precisely the factor loading from the
original specified model in Figure 8.1 that we intentionally eliminated to
illustrate the model modification process. Thus, the use of several modification criteria in our specification search was successful in obtaining the
original model in Figure 8.1.
The standardized estimates and selected model-fit indices for the final
model (Figure 8.1), where the modification in the Lozenges factor loading
is now included, and are shown in the second column of Table 8.2. All
of the parameters included are statistically significantly different from
zero (p < .05), and all of the fit indices now indicate an acceptable level of
fit with no additional model modifications indicated. Thus, we consider
this to be the final best fitting confirmatory factor model with our sample
variance-covariance data. The LISREL–SIMPLIS program is provided at
the end of the chapter for this model analysis.

8.7 Summary
This chapter discussed confirmatory factor models using the five basic building blocks from model specification through model modification. We began
by analyzing a confirmatory factor model that was misspecified (Figure 8.2)
and interpreted a few model-fit criteria where the fit of the model was
deemed not acceptable. We then used model modification criteria to modify
the model, which yielded the confirmatory factor model in Figure 8.1. This
confirmatory factor model was deemed to be our final best fitting model.
This final best fitting model can be further validated by testing the same
confirmatory factor model with other samples of data (see chapter 12).

Appendix: LISREL–SIMPLIS Confirmatory
Factor Model Program
Confirmatory Factor Model Figure 8.1
Observed Variables:
VISPERC CUBES LOZENGES PARCOMP SENCOMP WORDMEAN

Y102005.indb 174

3/22/10 3:26:03 PM

175

Confirmatory Factor Models

Covariance Matrix
49.064
9.810 22.182
27.928 14.482 81.863
9.117 2.515 5.013 12.196
10.610 3.389 3.605 13.217 26.645
19.166 6.954 13.716 18.868 28.502 58.817
Sample Size: 301
Latent Variables: Spatial Verbal
Relationships:
VISPERC - LOZENGES = Spatial
PARCOMP - WORDMEAN = Verbal
Print Residuals
Number of Decimals = 3
Path Diagram
End of problem
Exercise
1. Test the following hypothesized confirmatory factor model
(Figure 8.3) using the LISREL computer software program:
Sample Size: 3094
Observed variables:
Academic ability (Academic)
Self-concept (Concept)
Degree aspirations (Aspire)
Degree (Degree)
Occupational prestige (Prestige)
Income (Income)
Correlation matrix:
Academic
1.000
0.487
0.236
0.242
0.163
0.064

Concept

Aspire

Degree

Prestige

Income

1.000
0.206
0.179
0.090
0.040

1.000
0.253
0.125
0.025

1.000
0.481
0.106

1.000
0.136

1.000

Hypothesized CFA model: The CFA model indicates that the
first three observed variables measure the latent variable
Academic Motivation (Motivate) and the last three observed
variables measure the latent variable Socioeconomic Status
(SES). Motivate and SES are correlated.
Then modify the model to achieve a better model fit as shown in
Figure 8.4.

Y102005.indb 175

3/22/10 3:26:03 PM

176

A Beginner’s Guide to Structural Equation Modeling

Motivate

SES

Academic

err_ad

Concept

err_c

Aspire

err_l

Degree

err_d

Prestige

err_p

Income

err_i

Figure 8.3
Hypothesized CFA model for exercise.

Motivate

SES

Academic

err_ad

Concept

err_c

Aspire

err_l

Degree

err_d

Prestige

err_p

Income

err_i

Figure 8.4
Final CFA model for exercise.

Y102005.indb 176

3/22/10 3:26:04 PM

Confirmatory Factor Models

177

References
Cochran, W. G. (1968). Errors of measurement in statistics. Technometrics, 10,
637–666.
Comrey, A. L., & Lee, H. B. (1992). A first course in factor analysis. Hillsdale, NJ:
Lawrence Erlbaum.
Costello, A. B., & Osborne, J. (2005). Best practices in exploratory factor analysis: four recommendations for getting the most from your analysis. Practical
Assessment Research and Evaluation, 10(7), 1–9.
Fuller, W. A. (1987). Measurement error models. New York: Wiley.
Gorsuch, R. L. (1983). Factor analysis (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.
Harmon, H. H. (1976). Modern factor analysis (3rd ed., rev.). Chicago: University of
Chicago Press.
Holzinger, K. J., & Swineford, F. A. (1939). A study in factor analysis: The stability
of a bi-factor solution. (Supplementary Educational Monographs, No. 48).
Chicago, IL: University of Chicago, Department of Education.
Jöreskog, K. G., & Sörbom, D. (1993). LISREL 8: Structural equation modeling with the
SIMPLIS command language. Chicago, IL: Scientific Software International.

Y102005.indb 177

3/22/10 3:26:04 PM

9
Developing Structural Equation
Models: Part I

Key Concepts
Latent independent and dependent variables
Observed independent and dependent variables
Developing structural models with latent variables
Establishing relationships between latent variables
Covariance terms
The four-step approach to SEM modeling
Structural equation models have been developed in a number of academic
disciplines to substantiate and test theory. Structural equation models
have further helped to establish the relationships between latent variables
or constructs, given a theoretical perspective. The structural equation
modeling approach involves developing measurement models to define
latent variables and then establishing relationships or structural equation
models with the latent variables. The focus of this chapter is on providing researchers with a better understanding of how to develop structural
equation models. An attempt is made to minimize matrix and statistical
notation so that the reader can better understand the structural equation
modeling approach.
This chapter begins with a more extensive discussion of observed variables and latent variables, and then proceeds with sections on the measurement model, the structural model, variances and covariance terms,
and finally the two-step/four-step approaches to structural equation
modeling. Chapter 10 extends the development of SEM models in examining model specification, model identification, model estimation, model
testing, and model modification.

179

Y102005.indb 179

3/22/10 3:26:04 PM

180

A Beginner’s Guide to Structural Equation Modeling

9.1 Observed Variables and Latent Variables
In structural equation modeling, as in traditional statistics, we use X and
Y to denote the observed variables. We use X to refer to independent (or
predictor) variables and Y to refer to dependent (or criterion) variables; this
is the same in multiple regression, analysis of variance, and all general linear models. In structural equation modeling, however, we further define
latent independent variables using observed variables denoted by X and
latent dependent variables using observed variables denoted by Y. Latent
independent and dependent variables are created with observed variables
using confirmatory factor models discussed in the previous chapter.
There are two major types of variables in structural equation modeling:
observed (indicator) variables and latent (construct) variables. Latent variables are not directly observable or measured, rather they are observed or
measured indirectly, and hence they are inferred constructs based on what
observed variables we select to define the latent variable. For example,
intelligence is a latent variable and represents a psychological construct.
Intelligence cannot be directly observed, for example, through visual
inspection of an individual, and thus there is no single agreed upon definition for intelligence. However, intelligence can be indirectly measured
through observed or indicator variables, for example, specific IQ tests.
Observed or indicator variables are variables that are directly observable
or measured. For example, the Wechsler Intelligence Scale for Children–
Revised (WISC‑R) is an instrument commonly used to measure children’s
intelligence. The instrument represents one definition or measure of what
we mean by intelligence. Other researchers rely on other definitions or
observed measures, and thus on other instruments, for example, the
Stanford–Binet Intelligence Scale. Latent variables such as intelligence are
not directly observable or measured, but can be indirectly observed or
measured by using several observed (indicator) variables, for example, IQ
tests such as the WISC‑R and the Stanford–Binet Intelligence Scale.
Let us further examine the concept of latent variables as they are used
in structural equation models. Consider a basic structural equation model
in which we propose that a latent independent variable predicts a latent
dependent variable. For instance, Intelligence (latent independent variable)
is believed to predict subsequent Scholastic Achievement (latent dependent
variable), which could be depicted as
Intelligence → Achievement
Any latent variable that is predicted by other latent variables in a structural
equation model is known as a latent dependent variable. A latent dependent
variable, therefore, must have at least one arrow pointing to it from another

Y102005.indb 180

3/22/10 3:26:04 PM

Developing Structural Equation Models: Part I

181

latent variable, sometimes referred to as an endogenous latent variable. Any
latent variable that does not have an arrow pointing to it from another latent
variable is known as a latent independent variable, sometimes referred to as an
exogenous latent variable. As shown in the foregoing example, the latent independent variable Intelligence does not have any arrows pointing to it from
another latent variable. In our basic structural equation model, Intelligence
is the latent independent variable with no direct lines or arrows pointing to
it, and Achievement is the latent dependent variable because it has an arrow
pointing to it from Intelligence.
Consider adding a third latent variable to our basic structural equation model, such that Achievement is measured at two points in time. This
model would be depicted as follows:
Intelligence → Achievement1 → Achievement2.
Intelligence is still a latent independent variable. Achievement2 is clearly a
latent dependent variable because there is an arrow pointing to it from
Achievement1. However, there is an arrow pointing to Achievement1 from
Intelligence and another arrow pointing from Achievement1 to Achievement2.
This basic structural equation model indicates that Achievement1 is
predicted by Intelligence, but then Achievement1 predicts Achievement2.
Achievement1 is first a dependent latent variable and then an independent
latent variable. This type of structural equation model is possible and
illustrates indirect effects using latent variables. Achievement1 in this basic
structural equation model is a mediating latent variable. Our designation
of a latent variable as independent or dependent is therefore determined
by whether or not an arrow is drawn from one latent variable to another
latent variable. If no arrows point to a latent variable from another latent
variable in the structural equation model, then it is a latent independent
variable. If an arrow points to a latent variable from another latent variable
in the structural equation model, then it is a latent dependent variable.
Next, we consider the concept behind the observed or indicator variables. The latent independent variables are measured by observed independent variables via a confirmatory factor analysis measurement model
and traditionally denoted by X. The latent dependent variables are measured by observed dependent variables via a confirmatory factor analysis
measurement model and traditionally denoted by Y. Following our example, we might choose the WISC‑R and the Stanford–Binet Intelligence
Scale as observed independent measures of the latent independent variable Intelligence. We can denote these observed variables as X1 and X2. For
each of the achievement latent variables, we might choose the California
Achievement Test and the Metropolitan Achievement Test as our observed
dependent measures. If these measures are observed at two points in time,
then we can denote the observed variables of Achievement1 as Y1 and Y2,

Y102005.indb 181

3/22/10 3:26:05 PM

182

A Beginner’s Guide to Structural Equation Modeling

and those of Achievement2 as Y3 and Y4, respectively. In our SEM model,
each latent variable is measured by two observed variables.
What is the benefit of using more than one observed variable to assess a
latent variable? In using a single observed variable to assess a latent variable, we assume that no measurement error is associated with the measurement of that latent variable. In other words, it is assumed that the latent
variable is perfectly measured by the single observed variable, which is
typically not the case. We define measurement error quite generally here to
include errors due to reliability and validity issues (see chapter 8).
Reliability is concerned with the ability of a measure (score) to be consistent, commonly referred to as internal consistency, consistency over
time, and consistency using similar measures, to denote different types
of measurement error associated with observed variable scores. Would
Jamie’s score on the WISC‑R be about the same if measured today as compared with next week? Evidence of score reliability (consistency) could be
shown when a measure is given to the same group of individuals at two
points in time, and the scores are roughly equivalent. If only a single measure of a latent variable is used and it is not very reliable, then our latent
variable is not defined very well. If the reliability of a single observed
measure of a latent variable is known, then it is prudent to specify or
fix the measurement error in the SEM model. This is accomplished, for
example, in LISREL–SIMPLIS by setting the error variance of the single
variable. The error variance of a single variable is determined by the following formula:
Error Variance of X1 = (1 − reliability coefficient) (s2X1).
If the reliability of scores for X1 is .85 with a standard deviation of 5.00,
then the error variance would be computed as:
X1error variance = (1 − .85) (5.00)2 = .15 (25) = 3.75.
In the LISREL–SIMPLIS program, you would then add the following command line to set the error variance for X1:
Set the error variance of X1 to 3.75
Validity is concerned with the extent to which scores accurately define
a construct, which is score inference—commonly referred to as content,
factorial, convergent–divergent, and discriminant validity—to denote different types of score inference associated with observed scores. Our interest in validity is how well we can make an inference from the measured
scores to the latent variable; that is, how well do test scores indicate what
they purport to measure. Does Jamie’s score on the WISC‑R really measure

Y102005.indb 182

3/22/10 3:26:05 PM

Developing Structural Equation Models: Part I

183

her intelligence or something else, such as her height? Evidence of validity
is shown when two indicators of the same latent variable are substantially
correlated. For example, if WISC‑R and height were used as indicators
of the latent variable Intelligence, we would expect them to not be correlated. If only a single measure of a latent variable is used and the score is
not valid (for example, if height is used to measure intelligence), then our
latent variable is not well defined. Establishing the reliability of scores for
our observed variable helps in estimating the validity coefficients (factor
loadings) in our measurement model because score validity is limited by
the reliability of the observed variable scores; that is, the maximum validity coefficient is less than or equal to the product of the square roots of the
two reliability coefficients, ρXY ≤ ρXX ′ ρYY ′ .
If we selected WISC‑R and height as observed indicator variables for
the latent independent variable Intelligence it would certainly not be well
defined and would include measurement error. The selection of only
height as an observed indicator of Intelligence would increase the measurement error and poorly define the construct. Consequently, in selecting
observed variables to define a latent variable, we need to select observed
variables that show evidence of both score reliability and score validity
for the intended purpose of our study. Because of the inherent difficulty
involved in obtaining reliable and valid measures with a single observed
variable, we strongly encourage you to consider multiple indicator variables for each latent independent and dependent variable in the structural
equation model.
There are a few obvious exceptions to this recommendation, especially
when research indicates that only one observed variable is available. In this
case, you have no other choice than to define the latent variable using a
single observed variable or use the observed variable in a Multiple Indicator
Multiple Indicator Cause (MIMIC) model (see chapter 15). Jöreskog and Sörbom
(1993, p. 37, EX7A.SPL) provided the rationale and gave an example for setting
the error variance of a single observed variable (VERBINTM) in defining
the latent variable Verbint. The verbal intelligence test (VERBINTM) was
a fallible (unreliable) measure of the latent variable Verbint, and therefore
it was unreasonable to assume that the error variance was zero (perfectly
reliable). Consequently, the sample reliability coefficient for VERBINTM
was assumed to be rXX’ = .85 rather than 1.00 (perfectly reliable, zero error
variance). The assumed value of the reliability coefficient, hence designation of the error variance for VERBINTM, will affect parameter estimates as well as standard errors. A reliability coefficient of rXX’ = .85 for
VERBINTM is equivalent to an error variance of 0.15 times the variance of
VERBINTM (3.65)2. The assumed error variance of VERBINTM was computed as .15 (3.65)2 = 1.998.
If we can assume a reasonable reliability coefficient for an observed
variable, then multiplying the observed variable’s variance by 1 minus the

Y102005.indb 183

3/22/10 3:26:05 PM

184

A Beginner’s Guide to Structural Equation Modeling

reliability coefficient provides a reasonable estimate of error variance. In
the LISREL–SIMPLIS program EX7B.SPL, the error variance for the single
observed variable VERBINTM is accomplished by using the SET command as follows:
SET the error variance of VERBINTM to 1.998
Later in this chapter we show how measurement error is explicitly a part
of any structural equation model. The basic concept, however, is that
multiple observed variables used in defining either a latent independent variable or a latent dependent variable permit measurement error
to be estimated through structural equation modeling. This provides the
researcher with additional information about the measurement characteristics of the observed variables. When there is only a single observed
indicator of a latent variable, then measurement error cannot be estimated
through structural equation modeling, but can be fixed to a certain value.
Most SEM software programs, such as LISREL, permit the specification
of error variance for single or multiple variables, whether the values are
known or require our best guess. In the next two sections we discuss the
two approaches that make up structural equation modeling: the measurement model and the structural model.

9.2 Measurement Model
As previously mentioned, the researcher specifies the measurement model
to define the relationships between the latent variables and the observed
variables. The measurement model in SEM is a confirmatory factor model.
Using our previous example, the latent independent variable Intelligence is
measured by two observed variables, the WISC‑R and the Stanford–Binet
Intelligence Scale. Our other latent variables Achievement1 (dependent latent
variable) and Achievement2 (dependent latent variable) are each measured
by the same two observed variables, the California Achievement Test and
the Metropolitan Achievement Test, but at two different times. Both of these
observed variables are composite or scale scores from summing numerous
individual items. In chapter 8, we pointed out that individual items on an
instrument could be used to create a construct (latent variable); hence, confirming the unidimensionality of the construct, while taking into account
the observed variable score reliability and fit of the measurement model.
The use of many individual items rather than the composite score or item
parcels—that is, collections of individual items as the observed measures
of a latent variable—increases the degrees of freedom in the measurement
model and can cause problems in model fit. Measurement characteristics

Y102005.indb 184

3/22/10 3:26:05 PM

Developing Structural Equation Models: Part I

185

at the item level might be more appropriate for exploratory data reduction
methods than they are for SEM measurement models.
The researcher is typically interested in having the following questions
answered about the observed variables: To what extent are the observed
variables actually measuring the hypothesized latent variable; for example, how good is the California Achievement Test as a measure of achievement? Which observed variable is the best measure of a particular latent
variable; for example, is the California Achievement Test a better measure
of achievement than the Metropolitan Achievement Test? To what extent
are the observed variables actually measuring something other than the
hypothesized latent variable? For example, is the California Achievement
Test measuring something other than achievement, such as the quality of
education received? These types of questions need to be addressed when
creating the measurement models that define the latent variables.
In our measurement model example each latent variable is defined by
two indicator variables. The relationships between the observed variables and the latent variables are indicated by factor loadings. The factor
loadings provide us with information about the extent to which a given
observed variable is able to measure the latent variable (a squared factor loading indicates variable communality or amount of variance shared
with the factor). The factor loadings are referred to as validity coefficients
because multiplying the factor loading times the observed variable score
indicates how much of the observed variable score variance is valid (true
score). The observed variable measurement error is defined as that portion of the observed variable score that is measuring something other
than what the latent variable is hypothesized to measure. It serves as a
measure of error variance, and hence assesses the observed variable score
reliability. Measurement error could be the result of (a) an observed variable that is measuring some other latent variable, (b) unreliability, or (c) a
higher second order factor. For example, the California Achievement Test
may be measuring something besides achievement, or it may not yield
very reliable scores. Thus, we would like to know how much measurement error is associated with each observed variable.
In our measurement model there are six measurement equations, one
for each observed variable, which can be illustrated as follows:

California1 = function of Achievement1 + error

Metropolitan1 = function of Achievement1 + error
California2 = function of Achievement2 + error

Metropolitan2 = function of Achievement2 + error

WISC‑R = function of Intelligence + error
Stanford–Binet = function of Intelligence + error

Y102005.indb 185

3/22/10 3:26:06 PM

186

A Beginner’s Guide to Structural Equation Modeling

In the LISREL–SIMPLIS program, an explicit definition of the measurement model can be done by specifying measurement equations. One can
expand the variable labels in the measurement model equations using up
to eight characters; the labels are case-sensitive (upper and lower characters are recognized). The measurement model equations are specified,
using either the Relationships: or Paths: command (both methods are equivalent). For the Relationships: command, both the latent variables and the
observed variables can be written using eight-character variable names.
The observed variables are given on the left‑hand side of the equation with
spaces between the multiple observed variable names (Cal1, Metro1, Cal2,
Metro2, WISCR, and Stanford) and the latent variables on the right‑hand
side of the equation (Achieve1, Achieve2, and Intell). The LISREL–SIMPLIS
measurement equations follow where Achieve1 refers to Achievement1,
Intell refers to Intelligence, Achieve2 refers to Achievement2, Cal1 refers to
California1, Metro1 refers to Metropolitan1, Cal2 refers to California2, Metro2
refers to Metropolitan2, WISCR refers to WISC‑R, and Stanford refers to
Stanford–Binet). The command line for Relationships would be written as:
Relationships:
Cal1 Metro1 = Achieve1
Cal2 Metro2 = Achieve2
WISCR Stanford = Intell
For the Paths: command, the latent variables are depicted to the left of the
arrow and the observed variables to the right of the arrow with spaces
between the multiple observed variable names. The command line for
Paths in the following measurement equation would be written as:
Paths:
Achieve1 → Cal1 Metro1
Achieve2 → Cal2 Metro2
Intell → WISCR Stanford

9.3 Structural Model
In chapter 8 we discussed the rationale and process for specifying a
measurement model to indicate whether the latent variables are measured well, given a set of observed variables. If the latent variables

Y102005.indb 186

3/22/10 3:26:06 PM

Developing Structural Equation Models: Part I

187

(independent and dependent) are measured well, we then specify
a structural model to indicate how these latent variables are related.
The researcher specifies the structural model to allow for certain relationships among the latent variables depicted by the direction of the
arrows. In our example we hypothesized that intelligence and achievement are related in a specific way. We hypothesized that intelligence
predicts later achievement. The hypothesized structural model can now
be specified and tested to determine the extent to which these a priori
hypothesized relationships are supported by our sample variance–
covariance data; that is, Can intelligence predict achievement? Could
there be other latent variables that we need to consider to better predict
achievement? These types of questions are addressed when specifying
the structural model.
At this point we need to provide a more explicit definition of the structural model and a specific notational system for the latent variables under
consideration. Let us return to our previous example where we indicated
a specific hypothesized relationship for the latent variables:
Intelligence → Achievement1 → Achievement2 .
The hypothesized relationships for the latent variables indicate two latent
dependent variables, so there will be two structural equations. The first
equation should indicate that Achievement1 is predicted by Intelligence.
The second equation should indicate that Achievement2 is predicted by
Achievement1. These two equations can be illustrated as follows:
Achievement1 = structure coefficient1 * Intelligence + error
Achievement2 = structure coefficient2 *Achievement1 + error
These two equations specify the estimation of two structure coefficients
to indicate the magnitude (strength as well as statistical significance)
and direction (positive or negative) of the prediction. Each structural
equation also contains a prediction error or disturbance term that indicates the portion of the latent dependent variable that is not explained
or predicted by the other latent variables in that equation. In our example there are two structure coefficients, one for Intelligence predicting
Achievement1 and one for Achievement1 predicting Achievement2. Because
there are two structural equations, there are two prediction errors or
disturbances.
The LISREL–SIMPLIS command language permits an easy way to specify structural equations among the latent variables. The structural model
can be denoted in terms of either the Relationships: or Paths: commands

Y102005.indb 187

3/22/10 3:26:06 PM

188

A Beginner’s Guide to Structural Equation Modeling

(both methods are equivalent). For the Relationships: command, the latent
variables can be written using eight-character variable names with the
latent dependent variables on the left side of the equation (where Achieve1
refers to Achievement1, Intell refers to Intelligence, and Achieve2 refers to
Achievement2):
Relationships:
Achieve1 = Intell
Achieve2 = Achieve1
For the Paths: command, these latent dependent variables are to the right
of the arrow, as in the following structural equations:
Paths:
Intell → Achieve1
Achieve1 → Achieve2
You do not need to indicate the prediction error in LISREL–
SIMPLIS structural equations for either the Relationships: or Paths: commands because these are known to exist and automatically estimated by
the program.)

(Note :

The path diagram of the measurement and structural models for our
example is shown in Figure 9.1.

error

WISCR

Stanford

Cal1

Metro1

Cal2

Metro2

Intell

Achieve1

error

Achieve2

error

Figure 9.1
Achievement path model.

Y102005.indb 188

3/22/10 3:26:06 PM

Developing Structural Equation Models: Part I

189

9.4 Variances and Covariance Terms
In structural equation modeling, the term covariance structure analysis or
covariance structure modeling is often used because the estimation of factor
loadings and structure coefficients involves the decomposition of a sample
variance–covariance matrix. In this section we further explore the notion
of variance–covariance as it relates to observed and latent variables. There
are three different variance–covariance terms that we need to define and
understand. In the structural model there are two variance–covariance
terms to consider. First, there is a variance–covariance matrix of the latent
independent variables. This consists of the variances for each latent independent variable, as well as the covariance terms among them. Although
we are interested in the variances (the amount of variance associated with
the latent independent variable intelligence), the covariance terms may or
may not be part of our theoretical model. In our model there is only one
latent independent variable, so there is only one variance term and no
covariance term.
If we specified two latent independent variables in a different structural equation model, for example, Intelligence and Home Background, we
could include a covariance term for them. We would then be hypothesizing that Intelligence and Home Background are correlated or covary because
we believe that some common unmeasured latent variable is influencing
both of them. We could hypothesize that a latent variable not included in
the model, such as Parenting Ability, influences both Intelligence and Home
Background. In other words, Intelligence and Home Background co-vary, or
are correlated, because of their mutual influence from Parenting Ability,
which has not explicitly been included in the model (but which perhaps
could be included).
In the LISREL–SIMPLIS program, the variance term would automatically be given or implied in the output for the latent independent variable
Intelligence. A covariance term, if one existed, would also automatically
be given or implied in the output. If one desired the two latent independent variables, Intelligence and Home Background, to be uncorrelated or to
have a covariance of zero, then one would specify the following in the
LISREL–SIMPLIS program:
Set the Covariance between Intell and HomeBack to 0

The second set of variance–covariance terms that we need to define and
understand is in the covariance matrix of the structural equation model for
prediction errors. This consists of the variances for each structural equation prediction error (the amount of unexplained variance for each structural equation), as well as covariance terms among them. Although we are

Y102005.indb 189

3/22/10 3:26:07 PM

190

A Beginner’s Guide to Structural Equation Modeling

interested in the variances, the covariance terms may or may not be part of
our model. We could specify that two structural equation prediction errors
are correlated, perhaps because some unmeasured latent variable is leading to error in both equations. An example of this might be where Parental
Occupational Status (parent income) is not included as a latent variable in
a model where Children’s Education (in years) and Children’s Occupational
Status (income at age 30 years) are latent dependent variables. The structural equations for Children’s Education and Children’s Occupational Status
would then both contain structural equation prediction error due to the
omission of Parental Occupational Status. Because the same latent variable
was not included in both equations, we expect that the structural equation
prediction errors would be correlated. (Note: Our hypothesized structural
model does not contain any such covariance terms.)
In the LISREL–SIMPLIS program, the variance terms are automatically
included in the output for each structural equation. Because the covariance terms are assumed by the program to be set to zero, one must specify any covariance terms one wants estimated. A covariance term, if one
existed between Achievement1 and Achievement2, would be specified using
the following command:
Set the Error Covariance between Achieve1 and Achieve2 free

The third set of variance–covariance terms is from the measurement
model. Here, we need to define and understand the variances and covariance terms of the measurement errors. Although we are interested in
the variances (the amount of measurement error variance associated with
each observed variable), the covariance terms may or may not be part of
our model. We could hypothesize that the measurement errors for two
observed variables are correlated (known as correlated measurement error).
This might be expected in our example model where the indicators of the
latent variables Achievement1 and Achievement2 are the same—for example,
from using the California Achievement Test at two different times. We
might believe that the measurement error associated with the California
Achievement Test at Time 1 is related to the measurement error for the
California Achievement Test at Time 2.
In the LISREL–SIMPLIS program, the variance terms are automatically
specified in the program for each observed variable. Once again, the covariance terms are assumed by the program to be set to zero; so we must
specify any covariance term of interest and allow it to be estimated. A
covariance term, if one existed between the measurement errors for the
California Achievement Test at Times 1 and 2, would be specified using
the following command:
Set the Error Covariance between Cal1 and Cal2 free

Y102005.indb 190

3/22/10 3:26:07 PM

Developing Structural Equation Models: Part I

191

There is one final variance–covariance term that we need to mention,
and it really represents the ultimate variance–covariance for our combined measurement model and structural model. From the structure
coefficient parameters we estimate in the structural model, the factor
loadings in the measurement model(s), and all of the variance–covariance
terms, we generate an ultimate matrix of variance–covariance terms for
the overall SEM model. This variance–covariance matrix is implied by
the overall model and is denoted by Σ (see chapter 17 for a representation of all of these matrices). Our goal in structural equation modeling
is to estimate all of the parameters in the overall model and test the
overall fit of the model to the sample variance–covariance data. In short,
the parameters in our overall SEM model create an implied variance–
covariance matrix Σ from the sample variance–covariance matrix S,
which contains the sample variances and covariance terms among our
observed variables. We interpret our model-fit indices (see chapter 5) to
determine the level of model fit between Σ and S (closeness of the values in the variance–covariance matrix Σ implied by our hypothesized
model, and the sample variance–covariance matrix S given our sample
data). We also examine the magnitude (strength as well as statistical significance of parameter estimates) and the direction (positive or negative
coefficients) to provide a meaningful interpretation of our SEM model
results.

9.5 Two-Step/Four-Step Approach
James, Mulaik, and Brett (1982) proposed a two-step modeling approach
that emphasized the analysis of the two conceptually distinct latent variable models: measurement models and structural models. Anderson and
Gerbing (1988) described their approach by stating that the measurement
model provides an assessment of convergent and discriminant validity,
and the structural model provides an assessment of predictive validity.
Mulaik et al. (1989) also expanded the idea of assessing the fit of the structural equation model among latent variables (structural model) independently of assessing the fit of the observed variables to the latent variables
(measurement model). Their rationale was that even with few latent variables, most parameter estimates define the relationships of the observed
variables to the latent variables in the measurement model, rather than
the structural relationships of the latent variables themselves. Mulaik and
Millsap (2000) further elaborated a four-step approach discussed in chapter 5. Jöreskog and Sörbom (1993, p. 113) had earlier summarized many of
their thoughts by stating:

Y102005.indb 191

3/22/10 3:26:07 PM

192

A Beginner’s Guide to Structural Equation Modeling

The testing of the structural model, i.e., the testing of the initially
specified theory, may be meaningless unless it is first established that
the measurement model holds. If the chosen indicators for a construct
do not measure that construct, the specified theory must be modified
before it can be tested. Therefore, the measurement model should be
tested before the structural relationships are tested.

We have found it prudent to follow their advice. In the establishment of
measurement models, it is best to identify a few good indicators of each
latent variable with three or four indicators being recommended. In our
example, we intentionally used only a few indicators to define or measure
the latent variables to keep the model simple. We have also found that
when selecting only a few indicator variables, it is easier to check how
well each observed variable defines a latent variable—that is, to examine
the factor loadings, reliability coefficients, and the amount of latent variable variance explained. For example, rather than use individual items
as indicator variables, sum the items to form a total test score or a parcel
score (composite score or scale score). In addition, one can calculate the
reliability of the composite (scale) score and even consider fixing the value
of the relevant measurement error variance in the model (as described in
section 9.1), thus reducing the need to estimate one parameter. It is only
after latent variables are adequately defined (measured) that it makes
sense to examine latent variable relationships in a structural model. We
think a researcher with adequately measured latent variables is in a better position to establish a substantive, meaningful structural model, thus
supporting theory.

9.6 Summary
This chapter focused on how to develop structural equation models. We
began with a more detailed look at both observed and latent variables.
Next, we discussed the measurement and structural models. We extended
some of the basic concepts found in confirmatory factor models (measurement models) and regression/path models (structural models) to structural
equation modeling. We then described three types of variance–covariance
matrices typically utilized in structural equation models. The chapter concluded with a discussion of the popular two-step/four-step approaches
to structural equation modeling. In chapter 10 we extend our discussion
of the development of structural equation models by considering model
specification, model identification, model estimation, model testing, and
model modification, utilizing a more complex hypothesized theoretical
model.

Y102005.indb 192

3/22/10 3:26:07 PM

Developing Structural Equation Models: Part I

193

Exercises
1. Diagram two indicator variables X1 and X2 of a latent variable LV.
2. Diagram two observed variables X1 and X2 that predict a third
observed variable Y. X1, and X2 are correlated.
3. Diagram a latent independent variable LIV predicting a latent
dependent variable LDV.
4. Would you use a single indicator of a latent variable? Why or
why not?

References
Anderson, J. C., & Gerbing, D. W. (1988). Structural equation modeling in practice:
A review and recommended two-step approach. Psychological Bulletin, 103,
411–423.
James, L. R., Mulaik, S. A., & Brett, J. M. (1982). Causal analysis: Assumptions, models,
and data. Los Angeles, CA: Sage.
Jöreskog, K. G., & Sörbom, D. (1993). LISREL 8: Structural equation modeling with the
SIMPLIS command language. Chicago: Scientific Software International.
Mulaik, S. A., James, L. R., Alstine, J. V., Bennett, N., Lind, S., & Stilwell, C. D.
(1989). Evaluation of goodness-of-fit indices for structural equation models.
Psychological Bulletin, 105, 430–445.
Mulaik, S. A., & Millsap, R. E. (2000). Doing the four-step right. Structural Equation
Modeling, 7, 36–73.

Y102005.indb 193

3/22/10 3:26:07 PM

10
Developing Structural Equation
Models: Part II

Key Concepts
Factor loadings and measurement errors
Structure coefficients and prediction errors
Variance and covariance terms
Specification search
In chapter 9 we presented the basic framework for the development of
structural equation models. We focused on the measurement model, the
structural model, and the different variance–covariance terms. These
constitute the basic building blocks for analyzing and interpreting a
structural equation model. In this chapter we extend our discussion of the
development of structural equation models. We present a hypothesized
theoretical structural equation model and discuss issues related to model
specification, model identification, model estimation, model testing, and
model modification in the context of that example.

10.1 An Example
We hypothesized a structural equation model based on predicting educational achievement as a latent dependent variable. The structural model is
diagrammed in Figure 10.1 with four latent variables drawn as ellipses: two
latent independent variables, home background (Home) and Ability, and two
latent dependent variables, aspirations (Aspire) and achievement (Achieve).
Three of the latent variables are defined by using two indicator variables, and one latent variable, Home, is defined by using three indicator
variables in the measurement model. The indicator variables are depicted
using rectangles as follows: (a) for Home, family income (FamInc), father’s
education (FaEd) and mother’s education (MoEd); (b) for Ability, verbal
195

Y102005.indb 195

3/22/10 3:26:07 PM

196

error

A Beginner’s Guide to Structural Equation Modeling

FamInc

error
1

FaEd

error

MoEd

error

VerbAb

error

QuantAb

1
Home

error

OcAsp

error

VerbAch

error

QuantAch

error

Aspire

1
Ability

EdAsp

Achieve

error

Figure 10.1
Structural equation model of educational achievement.

ability (VerbAb) and quantitative ability (QuantAb); (c) for Aspire, educational aspiration (EdAsp) and occupational aspiration (OcAsp); and (d)
for Achieve, verbal achievement (VerbAch) and quantitative achievement
(QuantAch).
The measurement models for each latent variable identify which observed
variables define that particular latent variable. An arrow is drawn from the
latent variable to each of its observed indicator variables. For each arrow,
we understand that a factor loading will be computed. For example, the
observed measures of family income, father’s education and mother’s education define the latent variable Home, with each observed variable having a
factor loading estimated. Figure 10.1 has nine arrows going from the latent
variables drawn as ellipses to observed variables drawn as rectangles, thus
nine factor loadings will be estimated.
In Figure 10.1 each observed variable has a unique measurement error.
This is indicated by an arrow pointing to each observed variable and
shows that some portion of each observed variable is measuring something other than the hypothesized latent variable. For example, mother’s
education (MoEd) is hypothesized to define Home (home background), but
it may also be assessing other latent variables, a function of other variables
not in the model, random, or systematic error. The unique measurement
error is estimated for each observed variable, so there will be nine unique
measurement errors estimated. Each observed variable has a factor loading and a unique measurement error that forms an equation to compute
the latent variable score; for example,
MoEd = factor loading * Home + measurement error

Y102005.indb 196

3/22/10 3:26:08 PM

Developing Structural Equation Models: Part II

197

Our model diagram does not include any curved arrows for measurement
error, but this issue should be discussed. A curved arrow between two measurement error terms is possible and indicates that the measurement error
terms are correlated. Two measurement error terms could be correlated if
they share something in common, such as common method variance, or if
the same measure is being used at different points in time. For example,
quantitative ability (QuantAb) and quantitative achievement (QuantAch) may
have correlated measurement error terms, for example, because they represent two measures of quantitative skills. Correlated measurement error
terms may also exist for father’s education (FaEd) and mother’s education
(MoEd), for example, because using the same method of measurement, the
errors for one parent might be reflected in the other parent.
A straight arrow leading from a latent variable to a latent dependent variable designates that a structure coefficient is to be estimated. For example,
it was hypothesized that Home (home background) predicts Aspire (aspirations). The structure coefficients we want to estimate in our hypothesized
structural model come from a review of prior research and theory. In our
hypothesized structural model there are five structure coefficients we want
to estimate. Each latent dependent variable has one or more structure coefficients and a unique prediction error that forms an equation; for example,
Aspire = structure coefficient * Home + structure coefficient * Ability
+ prediction error
The prediction error for Aspire indicates that some portion of Aspire (aspiration) is not predicted by the latent independent variables Home and
Ability. There are two equations in our hypothesized structural model, so
we estimate two prediction errors, one for Aspire and one for Achieve:
Aspire = structure coefficient * Home + structure coefficient * Ability
+ prediction error
Achieve = structure coefficient * Home + structure coefficient * Ability
+ structure coefficient * Aspire + prediction error

10.2 Model Specification
Model specification is the first step in structural equation modeling (also
for regression models, path models, and confirmatory factor models). We
need theory because a set of observed variables can define a multitude
of different latent variables in a measurement model. In addition, many

Y102005.indb 197

3/22/10 3:26:08 PM

198

A Beginner’s Guide to Structural Equation Modeling

different structural models can be generated on the basis of different
hypothesized relationships among the latent variables.
In our theoretical structural equation model to predict Achieve (educational achievement) we used nine observed variables and hypothesized
four latent variables. Given this, many different measurement models and
structural models are possible. First, each observed variable can load on one
or more latent variables, so there could be nine or more possible factor loadings (up to 36 in our measurement model). Second, the two latent independent variables may or may not be correlated. Third, there may or may not be
correlations or covariance terms among the measurement errors, suggesting
there could be anywhere from zero to several possible correlated measurement error terms. Fourth, different structural models could be tested, so we
could have more than five or less than five structure coefficients in the different models. Finally, each structural equation has a prediction error—one
for each latent dependent variable—so we could have more or less prediction errors, and the prediction errors could be correlated.
How does a researcher determine which model is correct? We have
already learned that model specification is complicated, and we must
meet certain data conditions with the observed variables (see chapter 2).
Basically, structural equation modeling does not determine which model
to test; rather, it estimates the parameters in a model once that model has
been specified a priori by the researcher based on theoretical knowledge.
Consequently, theory plays a major role in formulating structural equation models and guides the researcher’s decision on which model(s) to
specify and test. Once again, we are reminded that model specification is
indeed the hardest part of structural equation modeling.
We used theory to formulate our measurement model and structural
model in predicting educational achievement, Achieve (Lomax, 1985). In the
measurement model there are nine equations, one for each observed variable.
From Figure 10.1, we formed the following nine measurement equations:
EdAsp = factor loading * Aspire + measurement error
OcAsp = factor loading * Aspire + measurement error

VerbAch = factor loading * Achieve + measurement error
QuantAch = factor loading * Achieve + measurement error
FamInc = factor loading * Home + measurement error
FaEd = factor loading * Home + measurement error

Y102005.indb 198

MoEd = factor loading * Home + measurement error

3/22/10 3:26:09 PM

Developing Structural Equation Models: Part II

199

VerbAb = factor loading * Ability + measurement error
QuantAb = factor loading * Ability + measurement error
Our latent variables are unobserved and have no definite scale of measurement (origin and unit of measurement are arbitrary). To define the
measurement model correctly, the origin and unit of measurement for
each latent variable must be defined. The origin of a latent variable is usually assumed to have a mean of 0. The unit of measurement (variance) of
a latent variable can be set using two different approaches. To compare
our factor loadings (interpret the parameter estimates), we need to define
a common unit of measurement for the latent variables. This is accomplished by setting a single observed variable factor loading to 1 for the
latent variable, for example, EdAsp = 1 * Aspire + measurement error. The
observed variable selected usually represents the best indicator of the
latent variable and is called a reference variable because all other observed
variables for that latent variable are interpreted in relation to its unit of
measurement. Another option is to assume that the latent variables have a
standardized unit of measurement and fix the latent variable variance to
1 (see Jöreskog & Sörbom, 1993, p. 173, 174).
In the LISREL–SIMPLIS command language (Jöreskog & Sörbom, 1993),
the measurement model equations are typically written using variable
names. In the Relationships: command, the observed variables are specified on the left‑hand side of the equation with spaces between the multiple
observed variable names and the latent variables on the right‑hand side.
The LISREL–SIMPLIS measurement equations are specified using variable names as follows:
Relationships:
EdAsp = 1*Aspire
OcAsp = Aspire
VerbAch = 1*Achieve
QuantAch = Achieve
FamInc = 1*Home
FaEd MoEd = Home
VerbAb = 1*Ability
QuantAb = Ability
The 1* notation in LISREL–SIMPLIS indicates parameters that are
fixed to 1.)

(Note :

Y102005.indb 199

3/22/10 3:26:09 PM

200

A Beginner’s Guide to Structural Equation Modeling

The equations for the structural model are
Aspire = structure coefficient * Home + structure coefficient * Ability
+ prediction error
Achieve = structure coefficient * Home + structure coefficient * Ability
+ structure coefficient * Aspire + prediction error
In LISREL–SIMPLIS, the structural model can be specified using a
Relationships: command. The latent variables can be written as eightcharacter variable names with either spaces or plus signs (+) used
between the latent variables. The prediction error terms for the two equations are assumed, so they are not included. The two structural equations
in LISREL–SIMPLIS are:
Relationships:
Aspire = Home Ability
Achieve = Aspire Home Ability
Finally, we must consider the three different types of variance–covariance
term terms. First, we check for variances and covariance terms among
the latent independent variables. For our model, there are separate variance terms for Home and Ability and a correlation term for the covariance
between Home and Ability. All of these parameter estimates are automatically specified in the LISREL–SIMPLIS program. Second, we check
for variances and covariance terms among the prediction errors. In our
model there are separate variance terms for each of the two structural
equations—that is, Aspire and Achieve—and no covariance term. These
variance terms are also automatically specified in the LISREL–SIMPLIS
program. Finally, we need to check for variance and covariance terms
among the measurement errors of the observed variables. In our measurement model equations there are nine variance terms for the observed
variables and no covariance terms. These are also automatically specified
in the LISREL–SIMPLIS program. Our careful attention to these details
assists in the specification of our structural equation model.

10.3 Model Identification
Once a structural equation model has been specified, the next step is to
determine whether the model is identified. In chapter 4 we pointed out that
the researcher must solve the identification problem prior to the estimation of

Y102005.indb 200

3/22/10 3:26:09 PM

Developing Structural Equation Models: Part II

201

parameters. For the identification problem, we ask the following question:
On the basis of the sample data contained in the sample variance–covariance
matrix S and the theoretical model implied by the population variance–
covariance matrix Σ, can a unique set of parameter estimates be found? For
the prediction of Achieve (educational achievement) specified in our theoretical model, we would like to know whether the factor loadings, measurement errors, structure coefficients, and prediction errors can be estimated
(identified). In our model we fixed certain parameters to resolve the origin
and unit of measurement problem (factor loading = 1) while leaving other
parameters free to be estimated. An example of a fixed parameter was setting the factor loading for FamInc (family income) on the latent independent
variable Home (home background) to 1. An example of a free parameter was
the factor loading for FaEd (father’s education) on Home (home background)
because it was not fixed, but rather free to be estimated.
We determine model identification by first checking the order condition. The number of free parameters to be estimated must be less than or
equal to the number of distinct values in the matrix S. A count of the free
parameters is as follows:
5 factor loadings (with 4 other factor loadings fixed to 1)
9 measurement error variances
0 measurement error covariance terms
2 latent independent variable variances
1 latent independent variable covariance
5 structure coefficients
2 equation prediction error variances
0 equation prediction error covariance terms
There are a total of 24 free parameters in our structural model that we want
to estimate. The number of distinct values in the matrix S is equal to
p (p + 1)/2 = 9 (9 + 1)/2 = 45,
where p is the number of observed variables in the sample variance–
covariance matrix. The number of values in S, 45, is greater than the number of free parameters, 24, so the model is probably identified, and we
should be able to estimate the number of free parameters that we specified. The degrees of freedom for our structural equation model is the difference between the number of distinct values in the matrix S and the
number of free parameters we want to estimate, df = 45 – 24 = 21. Thus,
according to the order condition, the model is overidentified, as there are
more values in S than parameters to be estimated.

Y102005.indb 201

3/22/10 3:26:09 PM

202

A Beginner’s Guide to Structural Equation Modeling

However, the order condition is only a necessary condition and is no
guarantee that the model is identified. Although the order condition is
easy to assess, other sufficient conditions are not, for example, the rank
condition. These other sufficient conditions require us to algebraically
determine whether each parameter in the model can be estimated from the
sample variance–covariance matrix S. According to the LISREL–SIMPLIS
computer program, which checks on identification through the rank test
and/or information matrix, the hypothesized structural equation model
for predicting Achieve (educational achievement) is identified.

10.4 Model Estimation
Once the identification problem has been resolved, the next step is to
estimate the parameters in the hypothesized structural equation model.
Once again, we can consider the traditional method of intuitively thinking about estimation by decomposing the variance–covariance (or correlation) matrix. The decomposition notion is that the original sample
variance–covariance (or correlation) matrix can be completely reproduced
if the relations among the observed variables are totally accounted for by
the theoretical model. If the model is not properly specified, the original
sample variance–covariance matrix will not be completely reproduced.
We now consider the estimation of the parameters for our hypothesized
structural model in Figure 10.1. The sample variance–covariance matrix
S is shown in Table 10.1 and the standardized residual matrix is shown
in Table 10.2. Our initial model was run in LISREL–SIMPLIS (LISREL–
SIMPLIS program in chapter Appendix).

Table 10.1
Sample Variance–Covariance Matrix for Example Data
Variable
1
2
3
4
5
6
7
8
9

Y102005.indb 202

EdAsp
OcAsp
VerbAch
QuantAch
FamInc
FaEd
MoEd
VerbAb
QuantAb

1.024
.792 1.077
1.027
.919 1.844
.756
.697 1.244 1.286
.567
.537
.876 .632 .852
.445
.424
.677 .526 .518
.434
.389
.635 .498 .475
.580
.564
.893 .716 .546
.491
.499
.888 .646 .508

.670
.545
.422
.389

.716
.373
.339

.851
.629 .871

3/22/10 3:26:09 PM

203

Developing Structural Equation Models: Part II

Table 10.2
Standardized Residual Matrix for Model 1
1

1. EdAsp
.000
2. OcAsp
.000
.000
3. VerbAch
1.420 −.797
.000
4. QuantAch −.776 −.363
.000
.000
5. FamInc
3.541 3.106 5.354 2.803
.000
6. FaEd
−2.247 −.578 −2.631 −.863 −2.809
.000
7. MoEd
−1.031 −1.034 −2.151 −.841 −3.240 6.338
.000
8. VerbAb
.877 1.956 −2.276 1.314 4.590 −.903 —2.144 .000
9. QuantAb −2.558
.185 1.820 −.574 3.473 —1.293 —2.366 .000 .000

The maximum likelihood estimates for the initial model are shown in
the first column of Table 10.3. All of the parameter estimates are within the
expected magnitude and direction based on previous research (Lomax,
1985). All of the parameter estimates are significantly different from
zero (p < .05), except the structure coefficient of Home predicting Achieve
(achievement) (standardized estimate = .139, t = 1.896, unstandardized estimate = .242). Because this structure coefficient is of substantive theoretical
interest, we will not remove it from the model. Aspire was statistically significantly predicted, R 2 = .612, and Achieve was statistically significantly,
predicted R 2 = .863, for both structural model equations. Home and Ability
latent variables were highly correlated, r = .728.

10.5 Model Testing
Model testing is the next crucial step in interpreting our results for the
hypothesized structural equation model. When the model-fit indices are
acceptable, the hypothesized model has been supported by the sample variance–covariance data. When the model-fit indices are not acceptable, we
usually attempt to modify the model by adding or deleting paths to achieve
a better model to data fit (see section 10.6).
For our initial model, we include several model-fit indices at the bottom
of Table 10.3 (see chapter 5). For the initial model, the c2 statistic, a measure of badness of fit, is equal to 58.85, 21 degrees of freedom, and p < .001.
Because the c2 value is statistically significant (p < .001) and is not close in
value to the number of degrees of freedom (recall NCP = 0, based on c2 −
df = 0), this model-fit index indicates that the initial model is unacceptable.
The root-mean-square error of approximation (RMSEA) is equal to .095,

Y102005.indb 203

3/22/10 3:26:10 PM

204

A Beginner’s Guide to Structural Equation Modeling

Table 10.3
Maximum Likelihood Estimates for Models 1 and 2
Estimates
OcAsp factor loading
QuantAch factor loading
FaEd factor loading
MoEd factor loading
QuantAb factor loading
Aspire -> Achieve coefficient
Home -> Aspire coefficient
Home -> Achieve coefficient
Ability -> Aspire coefficient
Ability -> Achieve
coefficient
Home variance
Ability variance
Home, Ability covariance
Aspire equation error
variance
Achieve equation error
variance
EdAsp error variance
OcAsp error variance
VerbAch error variance
QuantAch error variance
FamInc error variance
FaEd error variance
MoEd error variance
VerbAb error variance
QuantAb error variance
FaEd, MoEd error covariance
Goodness-of-fit indices:
c2
df
p value
GFI
AGFI
RMSR
RMSEA
a

Y102005.indb 204

Model 1

Model 2 (modified)

.917
.759
1.007
.964
.949
.548
.410
.242a
.590
.751

.918
.753
.782
.720
.949
.526
.506
.302a
.447
.685

.532
.663
.432
.335

.662
.663
.537
.319

.225

.228

.160
.351
.205
.342
.320
.130
.222
.188
.274
—

.161
.350
.193
.349
.190
.265
.373
.188
.274
.173

58.85
21
.000
.938
.868
.049
.095

18.60
20
.548
.980
.954
.015
.000

Estimates are not statistically significantly different from zero (p < .05). The c 2 values for
Model 1 and Model 2 can be checked for significance using Table A.4 in the Appendix.

3/22/10 3:26:10 PM

Developing Structural Equation Models: Part II

205

which is below the typical acceptable level of model fit (criterion RMSEA
< .08 or .05). The goodness-of-fit index (GFI) is .938 for the initial model,
which is around our acceptable range of model fit (criterion GFI > .95).
Finally, the adjusted goodness-of-fit index (AGFI) is .868 for this model,
not an acceptable level of fit (criterion AGFI > .95). From this particular set
of model-fit indices, we conclude that the hypothesized structural equation model is reasonable, but that some model modification might allow
us to achieve a more acceptable model to data fit. Model modification is
discussed in the next section.

10.6 Model Modification
The final step in structural equation modeling is to consider model modification to achieve a better model to data fit. If the hypothesized structural equation model has model-fit indices that are less than satisfactory,
a researcher typically performs a specification search to find a better fitting
model to the sample variance–covariance matrix. In chapter 4 we discussed the different procedures one can use in the specification search
process. For example, the researcher might eliminate parameters that are
not significantly different from zero and/or include additional parameters.
To eliminate parameters, the most commonly used procedure in LISREL–
SIMPLIS is to compare the t statistic for each parameter to a tabled t value,
for example, t > 1.96, a = .05, two-tailed test, or t > 2.58, a = .01, two-tailed
test (Table A.2) for statistical significance. To include additional parameters, the most commonly used techniques in LISREL–SIMPLIS are to
(a) select the highest modification index (MI; the expected value that c2
would decrease if such a parameter were to be included), and (b) select the
highest expected parameter change statistic (EPC; the approximate value
of the new parameter added to the model).
A researcher could also examine the residual matrix (or the more
useful standardized residual matrix) to obtain clues as to which original variances and covariance terms are not well accounted for by the
model (the residual matrix is the difference between the observed
variance–covariance terms in S and the corresponding model implied
(reproduced) variance–covariance terms in Σ). Large standardized residuals—for example, greater than 1.96 or 2.58—indicate that a particular
variable relationship is not well accounted for in the model.
For our hypothesized structural equation model, the original sample variance–covariance matrix is shown in Table 10.1 and the standardized residual
variance–covariance matrix is given in Table 10.2. The largest standardized
residual is for the relationship between FaEd (father’s education) and MoEd

Y102005.indb 205

3/22/10 3:26:10 PM

206

A Beginner’s Guide to Structural Equation Modeling

(mother’s education), which is 6.338. (Note: the t statistics do not suggest the
elimination of existing parameters, except one, from the initial model.) When
considering the addition of new parameters in the model, the largest modification index is for the measurement error covariance between FaEd (father’s
education) and MoEd (mother’s education), which is MI = 40.176. If we were
to estimate that parameter (correlation between FaEd and MoEd measurement errors), the expected parameter change would be EPC = 0.205.
In our specification search, the standardized residual and EPC values
indicated that something was wrong with how we specified the relationship between FaEd (father’s education) and MoEd (mother’s education),
because it was not specified well in the initial model. Consequently, we
decided to specify a measurement error covariance (correlation) between
FaEd (father’s education) and MoEd (mother’s education) because, upon
further reflection, there should be common method variance on measures
using the same scale with two different parents.
The ML estimates and selected model-fit indices for the modified model,
where the measurement error covariance is now included, are shown in
the second column of Table 10.3 and diagrammed in Figure 10.2. All of the
parameters are statistically significantly different from zero (p < .05), except
for the path between Home (home background) and Achieve (achievement),
but once again, for substantive theoretical reasons, we chose to leave this
relationship specified in the model. Our selected model-fit indices now all
indicate an acceptable level of fit, and a second specification search did not
result in any further recommended changes. Thus, we consider our modified model to be our final structural equation model for the prediction

error

FamInc

error
1

FaEd

error

MoEd

error

VerbAb

error

QuantAb

1
Home

error

OcAsp

error

VerbAch

error

QuantAch

error

Aspire

1
Ability

EdAsp

Achieve

error

Figure 10.2
Modified structural equation model of educational achievement.

Y102005.indb 206

3/22/10 3:26:11 PM

207

Developing Structural Equation Models: Part II

of educational achievement. Applying this structural equation model to
other samples of data will, we hope, provide further validation that this is
a theoretically meaningful structural model (see chapter 12).

10.7 Summary
This chapter completes the basic discussion of structural equation modeling we began in chapter 9. We hypothesized a structural equation model
to predict educational achievement and described it in further detail. We
followed the recommended steps a researcher should take in the structural equation modeling process, namely model specification, model
identification, model estimation, model testing, and finally model modification. We did not obtain acceptable model-fit indices with our initial
theoretical model, so we conducted a specification search. The specification search suggested adding a parameter estimate for the correlation
between the measurement error terms of father’s and mother’s education
level. The modified model resulted in acceptable model-fit indices, so this
was determined to be our best model to data fit. In chapter 11 we provide
suggestions and recommendations for how structural equation modeling
studies should be reported in the literature.

Appendix: LISREL–SIMPLIS Structural
Equation Model Program
Educational Achievement Example—Model 2
Observed variables: EdAsp OcAsp VerbAch
MoEd VerbAb QuantAb
Covariance matrix:
1.024
.792 1.077
1.027
.919 1.844
.756
.697 1.244 1.286
.567
.537
.876
.632 .852
.445
.424
.677
.526 .518 .670
.434
.389
.635
.498 .475 .545
.580
.564
.893
.716 .546 .422
.491
.499
.888
.646 .508 .389
Sample size: 200

Y102005.indb 207

Respecified
QuantAch FamInc FaEd

.716
.373
.339

.851
.629

.871

3/22/10 3:26:11 PM

208

A Beginner’s Guide to Structural Equation Modeling

Latent variables: ASPIRE ACHIEVE HOME ABILITY
Relationships:
EdAsp = 1*ASPIRE
OcAsp = ASPIRE
VerbAch = 1*ACHIEVE
QuantAch = ACHIEVE
FamInc = 1*HOME
FaEd MoEd = HOME
VerbAb = 1*ABILITY
QuantAb = ABILITY
ASPIRE = HOME ABILITY
ACHIEVE = ASPIRE HOME ABILITY
Let the error covariances of FaEd and MoEd correlate
Path diagram
End of problem
Exercise
1. Conduct the following structural equation model analysis
using the LISREL–SIMPLIS program:
Sample size = 500
Observed X variables:
ACT score (ACT)
College Grade Point Average (CGPA)
Company entry-level skills test score (ENTRY)
Observed Y variables:
Beginning salary (SALARY)
Current salary due to promotions (PROMO)
Latent dependent variable: Job Success (JOB)
Latent independent variable: Academic Success (ACAD)
Structural model:
ACAD -> JOB
Variance-covariance matrix:
ACT
CGPA
ENTRY
SALARY
PROMO

1.024
.792
.567
.445
.434

1.077
.537
.424
.389

.852
.518
.475

.670
.545

.716

References
Jöreskog, K. G., & Sörbom, D. (1993). LISREL 8: Structural equation modeling with the
SIMPLIS command language. Chicago, IL: Scientific Software International.
Lomax, R. G. (1985). A structural model of public and private schools. Journal of
Experimental Education, 53, 216–226.

Y102005.indb 208

3/22/10 3:26:11 PM

11
Reporting SEM Research: Guidelines
and Recommendations

Key Concepts
Theoretical models and data preparation
Sample matrix in SPSS or Excel
Model specification and identification
Model estimation and testing
Model modification and specification search
Breckler (1990) reviewed the personality and social psychology research
literature and found several shortcomings of structural equation modeling, namely that model-fit indices can be identical for a potentially
large number of models, that assumptions of multivariate normality are
required, that sample size affects results, and that cross-validation of
models was infrequently addressed or mentioned. Many of the studies
only reported a single model-fit index. Breckler concluded that there was
cause for concern in the reporting of structural equation modeling results.
Raykov, Tomer, and Nesselroade (1991) proposed guidelines for reporting
SEM results in the journal Psychology and Aging. Maxwell and Cole (1995)
offered some general tips for writing methodological articles, and Hoyle
and Panter (1995) published a chapter on reporting SEM research with an
emphasis on describing the results and what model-fit criteria to include.
The Publication Manual of the American Psychological Association (American
Psychological Association, 2001, pp. 161, 164–167, and 185) specifically
states that researchers should include the means, standard deviations,
and correlations of the entire set of variables so that others can replicate
and confirm the analysis, as well as provide example tables and figures
for reporting structural equation modeling research. Unfortunately, the
guidelines do not go far enough in outlining the basic information that
should be included to afford an evaluation of the research study and
some fundamental points that should be addressed when conducting
209

Y102005.indb 209

3/22/10 3:26:11 PM

210

A Beginner’s Guide to Structural Equation Modeling

SEM studies. A few other scholars have previously offered their advice,
as follows.
Boomsma (2000) discussed how to write a research paper when structural equation models were used in empirical research and how to decide
what information to report. His basic premise was that all information
necessary for someone else to replicate the analysis should be reported.
He provided recommendations along the lines of our basic steps in structural equation modeling, namely model specification, model identification, model estimation, model testing, and model modification. Boomsma
found that many studies lacked a theoretical foundation for the theoretical
model, gave a poor description of the model tested, provided no discussion
of the psychometric properties of the variables and level of measurement,
did not include sample data, and had a poor delineation or justification for
the model modification process. He pointed out how difficult it can be to
evaluate or judge the quality of published SEM research.
MacCallum and Austin (2000) provided an excellent survey of problems
in applications of SEM. Thompson (2000) provided guidance for conducting structural equation modeling by citing key issues and including the
following list of 10 commandments for good structural equation modeling behavior: (a) do not conclude that a model is the only model to fit the
data, (b) cross-validate any modified model with split-sample data or new
data, (c) test multiple rival models, (d) evaluate measurement models first,
then structural models, (e) evaluate models by fit, theory, and practical
concerns, (f) report multiple model-fit indices, (g) meet multivariate normality assumptions, (h) seek parsimonious models, (i) consider variable
scale of measurement and distribution, and (j) do not use small samples.
McDonald and Ringo Ho (2002) examined 41 of 100 articles in 13 psychological journals from 1995 to 1997. They stated that SEM researchers should give a
detailed justification of the SEM model tested along with alternative models,
account for identification, address nonnormality and missing data concerns,
and include a complete set of parameter estimates with standard errors, correlation matrix (and perhaps residuals), and goodness-of-fit indices.
We further elaborate several key issues in SEM. First, in structural equation model analyses several different types of sample data matrices can
be used (e.g., variance-covariance matrix, asymptotic variance-covariance
matrix, Pearson correlation matrix, or polyserial, polychoric, or tetrachoric
matrices). As previously described in chapter 3, the type of matrix used
depends on several factors such as nonnormality and type of variables.
A second issue concerns model identification, that is, the number of
distinct values in the sample variance-covariance matrix should equal or
exceed the number of free parameters estimated in the model (degrees of
freedom should not be negative for the model; the order condition) and
the rank of the matrix should yield a non-zero determinant value (the
rank condition). A researcher must also select from various parameter

Y102005.indb 210

3/22/10 3:26:11 PM

Reporting SEM Research: Guidelines and Recommendations

211

estimation techniques in model estimation, for example, unweighted least
squares, maximum likelihood, or generalized least squares estimation
under the assumption of multivariate normality, or asymptotically distribution-free estimation using ADF or CVM techniques when the multivariate normality assumption is not met. Obviously, many factors discussed
in chapters 2 and 3 affect multivariate normality.
A researcher should also be aware that equivalent models and alternative models may exist in an over-identified model (more distinct values in
the matrix than free parameters estimated), and rarely are we able to perfectly reproduce the sample variance-covariance matrix, given the implied
theoretical model. We use model-fit indices and specification searches to
obtain an acceptable model to data fit, given alternative models. Model-fit
statistics should guide our search for a better fitting model. Chapter 5 outlined different model-fit criteria depending on the focus of the research.
Under some situations, for example, use of large sample sizes, the chisquare values will be inflated leading to statistically significance, thus
erroneously implying a poor data to model fit. A more appropriate use
of the chi-square statistic in this situation would be to compare alternative models with the same sample data (nested models). The specification search process involves finding whether a variable should be added
(parameter estimated) or a variable deleted (parameter not estimated). A
researcher, when modifying an initial model, should make one modification at a time, that is, add or delete one parameter estimate, and give a
theoretical justification for the model change.
Ironically, structural equation modeling requires larger sample sizes
as models become more complex or the researcher desires to conduct
cross-validation with split samples. In traditional multivariate statistics
the rule of thumb is 20 subjects per variable (20:1). The rules of thumb
used in structural equation modeling vary from 100, 200, to 500 or more
subjects per study, depending on model complexity and cross-validation
requirements. Sample size and power are also important considerations
in structural equation modeling (see chapter 5). Finally, a two-step/fourstep approach is important because if measurement models do not fit the
observed variables, then relationships among the latent variables in structural models are not very meaningful.
We find the following checklist to be valuable when publishing SEM
research and hopefully journal editors will embrace the importance of
this information when published. Our checklist is:
1. Provide a review of literature that supports your theoretical model.
2. Provide the software program used along with the version.
3. Indicate the type of SEM model analysis (multi-level, structured
means, etc.).

Y102005.indb 211

3/22/10 3:26:11 PM

212

A Beginner’s Guide to Structural Equation Modeling

4. Include correlation matrix, sample size, means, and standard
deviations of variables.
5. Include a diagram of your theoretical model.
6. For interpretation of results, describe fit indices used and why; include
power and sample size determination; and effect size measure.
Our checklist is important because the SEM software, model, data, and
program will be archived in the journal. The power, sample size, and
effect size will permit future use in meta-analysis studies. Providing this
research information will also permit future cross-cultural research, multisample or multi-group comparisons, replication, or validation by others in
the research community because the analysis can be further examined.
We have made many of these same suggestions in our previous chapters, so our intentions in this chapter are to succinctly summarize guidelines and recommendations for SEM researchers.

11.1 Data Preparation
A researcher should begin a SEM research study with a rationale and purpose for the study, followed by a sound theoretical foundation of the measurement model and the structural model. This includes a discussion of
the latent variables and how they are defined in the measurement model.
The hypothesis should involve the testing of the structural model and/or
a difference between alternative models.
An applied SEM research study typically involves using sample data, in
contrast to a methodological simulation study. The sample matrix should
be described as to the type (augmented, asymptotic, covariance, or correlation), whether multivariate normality assumptions have been met,
the scale of measurement for the observed variables, and be related to
an appropriate estimation technique, for example, maximum likelihood.
Regression analysis, path analysis, factor analysis, and structural equation modeling all use data as input into a computer program (see SPSS
and Microsoft Excel examples at the end of the chapter). The SEM program should include the sample matrix, and for certain models, means
and standard deviations of the observed variables.
To show another way to input data, we can create special data file types
in SPSS by designating special rowtype_ and varname_ fields in the SPSS
Data Editor and entering variable names, an example of which follows
from the Holzinger and Swineford (1939) data previously presented in
chapter 8.

Y102005.indb 212

3/22/10 3:26:11 PM

Reporting SEM Research: Guidelines and Recommendations

213

We then enter the individual sample size, correlation coefficients, standard
deviations, and means for the girls’ data. We saved this file as girls_cor.sav.

A set of recommendations for data preparation is given in SEM Checklist
Box 11.1.

11.2 Model Specification
Model specification involves determining every relationship and parameter in the model that is of interest to the researcher. Moreover, the goal
of the researcher is to determine, as best possible, the theoretical model
that generates the sample variance–covariance matrix. If the theoretical
model is misspecified, it could yield biased parameter estimates; parameter estimates that are different from what they are in the true population model, that is, specification error. We do not typically know the true
population model, so bias in parameter estimates is generally attributed

Y102005.indb 213

3/22/10 3:26:12 PM

214

A Beginner’s Guide to Structural Equation Modeling

SEM Checklist Box 11.1—Data Preparation
1. Have you adequately described the population from which
the random sample data was drawn?
2. Did you report the measurement level and psychometric
properties (i.e., reliability and validity) of your variables?
3. Did you report the descriptive statistics on your variables?
4. Did you create a table with correlations, means and standard deviations?
5. Did you consider and treat any missing data (e.g., can result
in data analysis issues)? What was the sample size both
before and after treating the missing data?
6. Did you consider and treat any outliers (e.g., can affect sample statistics)?
7. Did you consider the range of values obtained for variables,
as restricted range of one or more variables can reduce the
magnitude of correlations?
8. Did you consider and treat any nonnormality of the data
(e.g., skewness and kurtosis, data transformations)?
9. Did you consider and treat any multicollinearity among the
variables?
10. Did you consider whether variables are linearly related,
which can reduce the magnitude of correlations?
11. Did you resolve any correlation attenuation (e.g., can result
in reduced magnitude of correlations and error messages)?
12. Did you take the measurement scale of the variables into
account when computing statistics such as means, standard
deviations, and correlations?
13. Did you specify the type of matrix used in the analysis (e.g.,
covariance, correlation (Pearson, polychoric, polyserial),
augmented moment, or asymptotic matrices)?
14. When using the correlation matrix, did you include standard deviations of the variables in order to obtain correct
estimates of standard errors for the parameter estimates?
15. How can others access your data and SEM program (e.g.,
appendix, Web site, email)?

Y102005.indb 214

3/22/10 3:26:12 PM

Reporting SEM Research: Guidelines and Recommendations

215

to specification error. The model should be developed from the available
theory and research in the substantive area. This should be the main purpose of the literature review.
Recommendations for model modification are provided in SEM
Checklist Box 11.2.
SEM Checklist Box 11.2—Model Specification
1. Did you provide a rationale and purpose for your study,
including why SEM rather than another statistical analysis
approach was required?
2. Did you describe your latent variables, thus providing a
substantive background to how they are measured?
3. Did you establish a sound theoretical basis for your measurement models and structural models?
4. Did you theoretically justify alternative models for comparison (e.g., nested models)?
5. Did you use a reasonable sample size, thus sufficient power
in testing your hypotheses?
6. Did you clearly state the hypotheses for testing the structural models?
7. Did you discuss the expected magnitude and direction of
expected parameter estimates?
8. Did you include a figure or diagram of your measurement
and structural models?
9. Have you described every free parameter in the models that
you want to estimate? In contrast, have you considered why
other parameters are not included in the models and/or why
you included constraints or fixed certain parameters?

11.3 Model Identification
In structural equation modeling it is crucial that the researcher resolve
the identification problem prior to the estimation of parameters in measurement models and/or structural models. In the identification problem, we
ask the following question: On the basis of the sample data contained in
the sample covariance matrix S, and the theoretical model implied by the
population covariance matrix Σ, can a unique set of parameter estimates
be found?
A set of recommendations for model identification includes the following shown in SEM Checklist Box 11.3.

Y102005.indb 215

3/22/10 3:26:12 PM

216

A Beginner’s Guide to Structural Equation Modeling

SEM Checklist Box 11.3—Model Identification
1. Did you specify the number of distinct values in your sample covariance matrix?
2. Did you indicate the number of free parameters to be
estimated?
3. Did you inform the reader that the order and/or rank condition was satisfied?
4. Did you report the number of degrees of freedom and
thereby the level of identification of the model?
5. How did you scale the latent variables (i.e., fix either one
factor loading per latent variable or the latent variable variances to 1.0)?
6. Did you avoid non-recursive models until identification
has been assured?
7. Did you utilize parsimonious models to assist with
identification?

11.4 Model Estimation
In model estimation we need to decide which estimation technique to select
for estimating the parameters in our measurement model and structural
model, that is, our estimates of the population parameters from sample
data. For example, we might choose the maximum likelihood estimation technique because we meet the multivariate normality assumption
(acceptable skewness and kurtosis); there are no missing data; no outliers; and continuous variable data. If the observed variables are interval
scaled and multivariate normal, then the ML estimates, standard errors
and chi‑square test are appropriate.
Our experience is that model estimation often does not work because
of messy data. In chapters 2 and 3 we outlined many of the factors that
can affect parameter estimation in general, and structural equation modeling specifically. Missing data, outliers, multicollinearity, and nonnormality of data distributions can seriously affect the estimation process and
often result in fatal error messages pertaining to Heywood variables (variables with negative variance), non-positive definite matrices (determinant
of matrix is zero), or failure to reach convergence (unable to compute a
final set of parameter estimates). SEM is a correlation research method
and all of the factors that affect correlation coefficients, the general linear
model (regression, path, and factor models), and statistics in general are

Y102005.indb 216

3/22/10 3:26:12 PM

Reporting SEM Research: Guidelines and Recommendations

217

compounded in structural equation modeling. Do not overlook the problems
caused by messy data!
Recommendations for model estimation are given in SEM Checklist
Box 11.4.
SEM Checklist Box 11.4—Model Estimation
1.
2.
3.
4.
5.

6.
7.

8.
9.
10.

What is the ratio of chi-square to the degrees of freedom?
What is the ratio of sample size to number of parameters?
Did you consider tests of parameter estimates?
Did you identify the estimation technique based on the type
of data matrix?
What estimation technique is appropriate for the distribution of the sample data (ML and GLS for multivariate normal data with small to moderate sample sizes; ADF or CVM
for non-normal, asymptotic covariance data, and WLS for
non-normal with large sample sizes)?
Did you encounter Heywood cases (negative variance), multicollinearity, or non-positive definite matrices?
Did you encounter and resolve any convergence problems
or inadmissible solution problems by using start values, setting the admissibility check off, using a larger sample size,
or using a different method of estimation?
Which SEM program and version did you use?
Did you report the R 2 values to indicate the fit of each separate equation?
Do parameter estimates have the expected magnitude and
direction?

11.5 Model Testing
Having provided the SEM program and sample data along with our measurement and structural models, anyone can check our results and verify our findings. In interpreting our measurement model and structural model, we
establish how well the data fit the models. In other words, we examine the
extent to which the theoretical model is supported by the sample data. In
model testing we consider model-fit indices for the fit of the entire model
and examine the specific tests for the statistical significance of individual
parameters in the model.
A set of recommendations for model testing includes the following as
shown in SEM Checklist Box 11.5.

Y102005.indb 217

3/22/10 3:26:13 PM

218

A Beginner’s Guide to Structural Equation Modeling

SEM Checklist Box 11.5—Model Testing
1. Did you report several model-fit indices (e.g., for a single
model: chi-square, df, GFI, NFI, RMSEA; for a nested model:
LR test, CFI, AIC; for cross-validation indices: CVI, ECVI;
and for parameter estimates, t values and standard errors)?
2. Did you specify separate measurement models and structural models?
3. Did you check for measurement invariance in the factor
loadings prior to testing between-group parameter estimates in the structural model?
4. Did you provide a table of estimates, standard errors, statistical significance (possibly including effect sizes and confidence
intervals)?

11.6 Model Modification
If the fit of an implied theoretical model is not acceptable, which is typically the case with an initial model, the next step is model modification and
subsequent evaluation of the new, modified model. Most of model modifications occur in the measurement model rather than the structural model.
Model modification occurs more in the measurement model because that
is where the main source of misspecification occurs and measurement
models are the foundation for the structural model.
After we are satisfied with our final best-fitting model, future research
should undertake model validation by replicating the study (using multiple
sample analysis, chapter 13), performing cross-validation (randomly splitting the sample and running the analysis on both sets of data), or bootstrapping the parameter estimates to determine the amount of bias. These
model validation topics are covered in chapter 12.
A set of recommendations for model modification is given in SEM
Checklist Box 11.6. Although not fully discussed until chapter 12, a set

SEM Checklist Box 11.6 —Model Modification
1. Did you compare alternative models or equivalent models?
2. Did you clearly indicate how you modified the initial
model?

Y102005.indb 218

3/22/10 3:26:13 PM

Reporting SEM Research: Guidelines and Recommendations

219

3. Did you provide a theoretical justification for the modified
model?
4. Did you add or delete one parameter at a time? What parameters were trimmed?
5. Did you provide parameter estimates and model-fit indices
for both the initial model and the modified model?
6. Did you report statistical significance of free parameters,
modification indices and expected change statistics of fixed
parameters, and residual information for all models?
7. How did you evaluate and select the best model?

SEM Checklist Box 11.7—Model Validation
1. Did you replicate your SEM model analysis using another
sample of data (e.g., conduct a multiple sample analysis)?
2. Did you cross-validate your SEM model by splitting your
original sample of data?
3. Did you use bootstrapping to determine the bias in your
parameter estimates?
of recommendations for model validation is provided in SEM Checklist
Box 11.7.

11.7 Summary
In this chapter we showed that model fit is a subjective approach that
requires substantive theory because there is no single best model (other
models may be equally plausible given the sample data and/or equivalent models). In structural equation modeling the researcher follows the
steps of model specification, identification, estimation, testing, and modification, so we advise the researcher to base measurement and structural
models on sound theory, utilize the two-step/four-step approach, and establish
measurement model fit and measurement invariance before model testing
the latent variables in the structural model. We also recommend that theoretical models need to be replicated, cross-validated, and/or bootstrapped to
determine the stability of the parameter estimates (see chapter 12). Finally,
we stated that researchers should include their SEM program, data, and
path diagram in any article. This permits a replication of the analysis and

Y102005.indb 219

3/22/10 3:26:13 PM

220

A Beginner’s Guide to Structural Equation Modeling

verification of the results. We do not advocate using specification searches
to find the best fitting model without having a theoretically justified reason for modifying the initial model. We further advocate using another
sample of data to validate that the modified model is a meaningful and
substantive theoretical structural model. Most importantly, we provide
the researcher with checklists to follow when doing structural equation
modeling. These checklists follow a logical progression from data preparation through model specification, identification, estimation, testing,
modification, and validation.
Exercise
1. Enter the following data in special matrix format in SPSS and
save as Fels_fem.sav. Use special variable names rowtype_ and
varname_ along with n, corr, stddev, and mean in these special
data sets.
N = 209
Correlation Matrix
Academic
Athletic
Attract
GPA
Height
Weight
Rating
s.d.
means

1.00
.43
.50
.49
.10
.04
.09
.16
.12

1.00
.48
.22
− .04
.02
.14
.07
.05

1.00
.32
−.03
−.16
.43
.49
.42

1.00
.18
−.10
.15
3.49
10.34

1.00
.34
−.16
2.91
.00

1.00
−.27
19.32
94.13

1.00
1.01
2.65

References
American Psychological Association (2001). Publication manual of the American
Psychological Association (5th ed.). Washington, DC: Author.
Boomsma, A. (2000). Reporting analyses of covariance structure. Structural Equation
Modeling, 7, 461–483.
Breckler, S. J. (1990). Applications of covariance structure modeling in psychology:
Cause for concern? Psychological Bulletin, 107, 260–273.
Holzinger, K. J., & Swineford, F. A. (1939). A study in factor analysis: The stability
of a bi-factor solution. (Supplementary Educational Monographs, No. 48).
Chicago: University of Chicago, Department of Education.

Y102005.indb 220

3/22/10 3:26:13 PM

Reporting SEM Research: Guidelines and Recommendations

221

Hoyle, R. H., & Panter, A. T. (1995). Writing about structural equation models. In R.
H. Hoyle (Ed.), Structural equation modeling: Concepts, issues, and applications
(pp. 158–176). Thousand Oaks, CA: Sage.
MacCallum, R. C., & Austin, J. T. (2000). Applications of structural equation modeling in psychological research. Annual Review of Psychology, 51, 201–226.
Maxwell, S. E., & Cole, D. A. (1995). Tips for writing (and reading) methodological
articles. Psychological Bulletin, 118, 193–198.
McDonald, R. P., & Ringo Ho, M. (2002). Principles and practice in reporting structural equation analyses. Psychological Methods, 7, 64–82.
Raykov, T., Tomer, A., & Nesselroade, J. R. (1991). Reporting structural equation modeling results in Psychology and Aging: Some proposed guidelines.
Psychology and Aging, 6, 499–533.
Thompson, B. (2000). Ten commandments of structural equation modeling. In L.
Grimm & P. Yarnold (Eds.), Reading and understanding more multivariate statistics (pp. 261–284). Washington, DC: American Psychological Association.

Y102005.indb 221

3/22/10 3:26:13 PM

12
Model Validation

Key Concepts
Replication: multiple samples
Cross validation: randomly split subsamples
Cross validation indexes: ECVI, CVI, and MECVI
Bootstrap via LISREL and PRELIS
Bootstrap via program menu
In previous chapters we learned about the basics of structural equation modeling using the following steps: model specification, identification, estimation, testing, and modification. In this chapter we consider a selection of
topics related to model validation. However, our discussion only scratches
the surface of these approaches in structural equation modeling, so you
should check out the references in this chapter for more information.
We begin by presenting the topic of replication, which uses multiple
samples. In our first example, the validation of a theoretical confirmatory factor model using two samples of data is presented. Cross validation is presented next, where a larger sample is randomly split into two
subsamples. Then, we present the basics of how to determine the stability
of parameter estimates using the bootstrap method. Ideally, a researcher
should seek model validation with additional samples of data (replication).
The other methods are not as rigorous, but in the absence of replication,
provide evidence of model validity—that is, the viability of the theoretical
framework suggested by the measurement and/or structural models.

12.1 Multiple Samples
A nice feature of structural equation modeling, although not frequently
used, is the possibility of studying a theoretical model and then validating
it using one or more additional samples of data. Theoretical models can
223

Y102005.indb 223

3/22/10 3:26:14 PM

224

A Beginner’s Guide to Structural Equation Modeling

therefore be examined across samples to determine the degree of invariance in fit indices, parameter estimates, and standard errors.
SEM also permits the use of multiple samples in the analysis of quasiexperimental, experimental, cross‑sectional, and/or longitudinal data.
With multiple samples it is possible to estimate separately the parameters for each independent sample, to test whether specified parameters or
parameter matrices are equivalent across the samples (that is, for any of
the parameters in the measurement and/or structural equation models),
or to test whether there are sample mean differences for the indicator variables and/or for any of the structural equations.
We can obviously estimate parameters in each sample of data separately. We would fit a theoretical model to the first sample of data and
then apply the model to the other samples of data. It is possible that a
confirmatory factor model will fit all samples of data (multiple samples),
indicating measurement invariance, and yet have different values for
error covariance, factor loadings, or factor correlations. We can also statistically determine whether certain specified parameters or parameter
matrices are equivalent across samples of data. For instance, one may
be interested in whether factor loadings and factor correlations are stable across random samples of data applied to a theoretical model. We
could also randomly split a large national sample of data into several
subsamples.
SEM also permits the testing of the equivalence of matrices or parameter estimates across several samples taken randomly from a population.
A researcher indicates the specific hypothesis to be tested, for example,
equal factor loadings and factor correlation. For a measurement model,
we could test whether the factor loadings are equal across the samples,
or whether the factor variances and covariance terms are equal across the
samples, or even whether the unique error variances and covariance terms
are equal across samples. For a structural model, we could test whether
the structure coefficients are equal across the samples. For a combined
structural equation model, all parameters in the entire model are tested
for equivalence across the samples. Obviously, in this instance both the
covariance matrix and the coefficients are tested for equality across the
samples, lending itself to a more complex model requiring adequate sample size and power.
In this chapter, we present four models: Model A with all parameters
invariant; Model B with only error variance and factor correlation invariant; Model C with only factor correlation invariant; and finally Model
D with factor loadings and factor correlation invariant. These examples
should give you a better understanding of how different model attributes
can be tested using multiple samples.
We now demonstrate how to conduct these multiple sample analyses
in LISREL–SIMPLIS using the example in Jöreskog and Sörbom (1996c,

Y102005.indb 224

3/22/10 3:26:14 PM

225

Model Validation

err_v40

VERBAL40
Verbal

err_v50

VERBAL50

err_m35

MATH35
Math

err_m25

MATH25

Figure 12.1
Path Diagram for SAT Verbal and Math. (From Jöreskog, K. G., & Sörbom, D., 1993.
Bootstrapping and Monte Carlo experimenting with PRELIS2 and LISREL8. Chicago: Scientific
Software International; example 10, p. 52.)

example 10, p. 52) that indicated two samples of data on candidates who
took the Scholastic Aptitude Test (SAT) in 1971 (Figure 12.1).
In LISREL–SIMPLIS, measurement and/or structural models can
be specified and tested across samples of data for model validation.
The LISREL–SIMPLIS program stacks separate programs, but does not
require that the observed variables, latent variables, and equations be
repeated in each program when the parameters in the theoretical model
are assumed identical in subsequent individual programs. The second
and subsequent individual programs only need to include their sample
size and variance–covariance matrix. Each individual program, however, must be designated by use of the special command, Group 1: and
Group 2: etc. (Note: Do not use Sample 1: and Sample 2: etc. to designate
the individual programs).
The computer output indicates results for each individual program
with chi-square contributions to the overall global chi-square value. Each
individual program outputs a chi-square value that sums to the global
chi-square value. A percent contribution to the global chi-square value is
also indicated for each individual program. In a multiple sample analysis,
the global chi-square is a measure of fit in all samples to the theoretical model. (Note: Individual sample parameter values can be obtained by
including the latent variable and equation statements in each subsequent
stacked program.)
LISREL–SIMPLIS generally requires running different multiple sample program models to determine which parameters are different or

Y102005.indb 225

3/22/10 3:26:14 PM

226

A Beginner’s Guide to Structural Equation Modeling

similar among factor loadings, error variances, and factor correlations. In
LISREL–SIMPLIS, Model A (EX10A.SPL) tests the equality of all parameters across both samples (factor loadings, error variances, and factor
correlation). Model B (EX10B.SPL) allows the factor loadings to be different, but maintains equal error variances and factor correlation. Model
C (EX10C.SPL) allows the factor loadings and error variances to be different, but maintains equal factor correlation across the two samples.
Finally, Model D (EX10D.SPL) specifies that the factor loadings and the
factor correlation are the same for both samples with the error variances
different.
The LISREL–SIMPLIS Model A program (EX10A.SPL), which tests
equality of all parameters (invariant) across both samples, is:
Group 1: Testing Equality of all model parameters
Model A: Factor Loadings, Correlation, Error Variances Invariant

Observed Variables: VERBAL40 VERBAL50 MATH35 MATH25
Covariance Matrix from File EX10.COV
Sample Size = 865
Latent Variables: Verbal Math
Relationships:
VERBAL40 VERBAL50 = Verbal
MATH35 MATH25 = Math
Group 2: Testing Equality of all model parameters
Covariance Matrix from File EX10.COV
Sample Size = 900
Path diagram
End of problem

12.1.1 Model A Computer Output
Group 1:
Contribution to Chi-Square
Percentage Contribution to
Group 2:
Contribution to Chi-Square
Percentage Contribution to
Global Goodness-of-Fit Statistics
Degrees of Freedom = 11
Minimum Fit Function Chi-Square =

= 19.16
Chi-Square = 54.92
= 15.73
Chi-Square = 45.08
34.89 (P = 0.00026)

The global chi-square is significant, so having all parameters equal (invariant) is not a tenable solution. Some parameters are therefore different in
the two samples.
The LISREL–SIMPLIS Model B program (EX10B.SPL), which tests differences in factor loadings with equal error variance and factor correlation, is:

Y102005.indb 226

3/22/10 3:26:14 PM

227

Model Validation

Group 1: Testing Equality Of Factor Structures
Model B : Factor Correlation and Error Variances Invariant
Observed Variables: VERBAL40 VERBAL50 MATH35 MATH25
Covariance Matrix from File EX10.COV
Sample Size = 865
Latent Variables: Verbal Math
Relationships:
VERBAL40 VERBAL50 = Verbal
MATH35 MATH25 = Math
Group 2: Testing Equality Of Factor Correlations
Covariance Matrix from File EX10.COV
Sample Size = 900
Relationships:
VERBAL40 VERBAL50 = Verbal
MATH35 MATH25 = Math
Path diagram
End of problem

12.1.2 Model B Computer Output
Group 1:
Contribution to Chi-Square
Percentage Contribution to
Group 2:
Contribution to Chi-Square
Percentage Contribution to
Global Goodness-of-Fit Statistics
Degrees of Freedom = 7
Minimum Fit Function Chi-Square =

= 15.62
Chi-Square = 52.65
= 14.05
Chi-Square = 47.35
29.67 (P = 0.00011)

The global chi-square was significant, which indicates that equal error
variance and equal factor correlation are not tenable results with factor
loadings being different.
The LISREL–SIMPLIS Model C program (EX10C.SPL), which tests differences in factor loadings and error variance, but equal in factor correlation, is:
Group 1: Testing Equality of Factor Structures
Model C: Factor Correlation Invariant
Observed Variables: VERBAL40 VERBAL50 MATH35 MATH25
Covariance Matrix from File EX10.COV
Sample Size = 865
Latent Variables: Verbal Math
Relationships:
VERBAL40 VERBAL50 = Verbal
MATH35 MATH25 = Math
Group 2: Testing Equality of Factor Correlations
Covariance Matrix from File EX10.COV
Sample Size = 900

Y102005.indb 227

3/22/10 3:26:15 PM

228

A Beginner’s Guide to Structural Equation Modeling

Relationships:
VERBAL40 VERBAL50 = Verbal
MATH35 MATH25 = Math
Set the Error Variances of VERBAL40 - MATH25 free
Path diagram
End of problem

12.1.3 Model C Computer Output
Group 1:
Contribution to Chi-Square
Percentage Contribution to
Group 2:
Contribution to Chi-Square
Percentage Contribution to
Global Goodness-of-Fit Statistics
Degrees of Freedom = 3
Minimum Fit Function Chi-Square =

= 2.21
Chi-Square = 55.02
= 1.81
Chi-Square = 44.98
4.03 (P = 0.26)

The factor correlation was r = .76 for both samples. This is tenable, given
the nonsignificant global chi-square statistic (c2 = 4.03, df = 3, p = .26).
The LISREL–SIMPLIS Model D program (EX10D.SPL), which tests factor
loadings and factor correlation the same (invariant), but allows for differences in error variance is:
Group 1: Testing Equality of Factor Structures
Model D: Factor Loadings and Factor Correlation Invariant
Observed Variables: VERBAL40 VERBAL50 MATH35 MATH25
Covariance Matrix
63.382
70.984 110.237
41.710 52.747 60.584
30.218 37.489 36.392 32.295
Sample Size = 865
Latent Variables: Verbal Math
Relationships:
VERBAL40 VERBAL50 = Verbal
MATH35 MATH25 = Math
Group 2: Testing Equality of Factor Correlations
Covariance Matrix
67.898
72.301 107.330
40.549 55.347 63.203
28.976 38.896 39.261 35.403
Sample Size = 900
Set the Error Variances of VERBAL40 - MATH25 free
Path diagram
End of problem

Y102005.indb 228

3/22/10 3:26:15 PM

229

Model Validation

12.1.4 Model D Computer Output
Group 1:
Contribution to Chi-Square
Percentage Contribution to
Group 2:
Contribution to Chi-Square
Percentage Contribution to
Global Goodness-of-Fit Statistics
Degrees of Freedom = 7
Minimum Fit Function Chi-Square =

= 5.48
Chi-Square = 50.40
= 5.39
Chi-Square = 49.60
10.87 (P = 0.14)

The global chi-square indicated a good fit of the measurement model
across both samples of data. Therefore, equal factor loadings and factor
correlation with unequal error variances is tenable. Error variances would
typically be different in a measurement model, so assuming equal factor
loadings and factor correlation was theoretically reasonable to test.
12.1.5 Summary
Although the multiple sample programs provide the individual and
global chi-square values, the researcher should consider creating a table
with the parameter values and standard errors. This would provide an
easier comparison of the intended parameter estimates that were modeled
in the different programs.
More complex model comparisons are possible. For example, we could
test the equality of both factor loadings and factor correlations across
three samples of data. Many different measurement and structural models
using the multiple sample approach are possible and have been illustrated
in journal articles, software manuals, and books. The interested reader is
referred to Jöreskog and Sörbom (1993), Muthén (1987) and Bentler and
Wu (2002), as well as books by Hayduk (1987) and Bollen (1989), for more
details on running these various multiple sample models. Other empirical
examples using multiple-sample models are given by Lomax (1983, 1985),
Cole and Maxwell (1985), Faulbaum (1987), and McArdle and Epstein
(1987). A suggested strategy for testing models in the multiple sample case
is also given by Lomax (1983).

12.2 Cross Validation
The replication of a study with a second set of data is often prohibitive
given the time, money, or resources available. An alternative is to randomly split an original sample, given that the sample size is sufficient,

Y102005.indb 229

3/22/10 3:26:15 PM

230

A Beginner’s Guide to Structural Equation Modeling

and run the SEM analysis on one set of data while using the other in a
multiple-sample analysis to compare the results. Cudeck and Browne
(1983) created a split sample cross-validation index (CVI), while Browne
and Cudeck (1989, 1993) developed a single sample cross-validation
(ECVI) and further explained CVI and ECVI in structural equation modeling. Except for a constant scale factor, ECVI is similar to the AIC index
[(1/n)* (AIC)]. Arbuckle and Wothke (1999, p. 406) also report MECVI,
which, except for a scale factor, is similar to BCC [(1/n) * (BCC)]. The
Browne–Cudeck criterion (BCC) imposes a slightly greater penalty for
model complexity than AIC, and is a fit index developed specifically
for the analysis of moment structures. These fit indices are intended for
model comparisons, and thus indicate badness of fit; with simple models
that fit well receiving low values and poorly fitting models receiving
high values.
12.2.1 ECVI
Browne and Cudeck (1989) proposed a single-sample expected crossvalidation index (ECVI) for comparing alternative models using only
one sample of data. The alternative model that results in the smallest
ECVI value should be the most stable in the population. The ECVI is
a function of chi-square and degrees of freedom. It is computed in
LISREL as ECVI = (c/n) + 2(p/n), where c is the chi-square value for
the overall fitted model, p is the number of independent parameters
estimated, and n = N − 1 (sample size). Alternatively, ECVI can be
reported as similar to the Akaike Information Criterion, except for a
scale factor—that is, (1/n) * AIC, where n = N – r (N = sample size; r =
number of groups). Browne and Cudeck (1989, 1993) also provided a
confidence interval for ECVI. The 90% lower and upper limits(cL ; cU) =
[(d L + d + 2q)/n; (d U + d + 2q)/n], where cL = lower limit, cU = upper limit, d L
= parameter estimate for lower limit, d U = parameter estimate for upper
limit, d = degrees of freedom, and q = the number of parameters. When
sample size is small, it is important to compare the confidence intervals
of the ECVI for the alternative competing models. The ECVI is also
not very useful for choosing a parsimonious model when the sample
size is large. In this instance, we recommend one of the parsimonious
model-fit indices and/or the comparative fit index if comparing alternative models (see chapter 5).
Bandalos (1993), in a simulation study, further examined the use of the
one-sample expected cross-validation index and found it to be quite accurate in confirmatory factor models. Other research also indicated that the
one-sample expected cross validation index yielded highly similar results
to those of the two-sample approach (Benson & Bandalos, 1992; Benson &

Y102005.indb 230

3/22/10 3:26:15 PM

Model Validation

231

El-Zahhar, 1994; Benson, Moulin-Julian, Schwarzer, Seipp, & El-Zahhar,
1992).
The ECVI is routinely printed among the fit indices reported by LISREL–
SIMPLIS. We used our previous multiple-sample programs in LISREL–
SIMPLIS, but this time ran them separately to obtain the ECVI values. The
ECVI for sample one was close to zero, indicating a measurement model
that would be expected to cross-validate; likewise similar findings were
reported for the second sample of data. The confidence intervals around
ECVI in both programs further supported that ECVI would probably
range between .02 and .03 for this model. (Note: We would not interpret
the ECVI in the multiple-sample model.)
LISREL–SIMPLIS ECVI Output
Sample 1
Expected Cross-Validation Index (ECVI) = 0.021
90 Percent Confidence Interval for ECVI = (0.019 ; 0.028)
ECVI for Saturated Model = 0.011
ECVI for Independence Model = 3.05

Sample 2
Expected Cross-Validation Index (ECVI) = 0.021
90 Percent Confidence Interval for ECVI = (0.021 ; 0.029)
ECVI for Saturated Model = 0.022
ECVI for Independence Model = 3.00

The AIC and BCC values can be computed to show the scale factor relationship to ECVI. AIC = c2 + 2q = 1.3 + 2(9) = 19.3, that is, reported as 19.255
for the first sample, where q = number of parameters in the model. AIC =
c2 + 2q = .9 + 2(9) = 18.922 for the second sample. ECVI = [1/(N – r)](AIC) =
[1/(865 – 2)](19.255) = .022 for sample 1 and ECVI = [1/(N – r)] (AIC) = [1/
(900 – 2)](18.922) = .021 for sample 2. N is the sample size in each group and
r is the number of groups. MECVI doesn’t apply in this model analysis,
but is computed as: [1/(N – r)] (BCC) or [1/(865 – 2)] (19.36) and [1/(900 − 2)]
(19.023), respectively.
12.2.2 CVI
Cudeck and Browne (1983) also proposed a cross-validation index (CVI)
for covariance structure analysis that incorporated splitting a sample into
two subsamples. Subsample A is used as a calibration sample, and subsample B is used as the validation sample. The model implied (reproduced)

Y102005.indb 231

3/22/10 3:26:15 PM

232

A Beginner’s Guide to Structural Equation Modeling

covariance matrix, Σa, from the calibration sample is then compared with
the covariance matrix derived from Subsample B, Sb. A CVI value near
zero indicates that the model cross-validates or is the same in the two
subsamples. The cross validation index is denoted as CVI = F(Sb, Σa). The
choice among alternative models can also be based on the model that
yields the smallest CVI value. One could further double-cross-validate by
using Subsample B as the calibration sample and Subsample A as the
validation sample. In this instance, the cross validation index is denoted
as CVI = F(Sa, Σb). If the same model holds regardless of which subsample
is used as the calibration sample, greater confidence in the model validity is achieved. An obvious drawback to splitting a sample into two subsamples is that sufficient subsample sizes may not exist to provide stable
parameter estimates. Obviously, this approach requires an initial large
sample that can be randomly split into two subsamples of equal and sufficient size.
The CVI can be computed using LISREL–SIMPLIS command language,
but requires two programs with randomly split data and the cross-validate
command. In the following example, two LISREL–SIMPLIS programs are
run to compute the CVI. The first program reads in the covariance matrix
of the calibration sample (Sa), then generates and saves the model implied
covariance matrix, Σa. The second program uses the covariance matrix of
Subsample B and then outputs the CVI value. The CVI cross validation
example involved randomly splitting an original sample of size 400 and
calculating two separate covariance matrices.
Program One Calibration Sample
Observed Variables: X1 X2 X3
Covariance Matrix
5.86
3.12 3.32
35.28 23.85 622.09
Latent Variables: Factor1
Relationships:
X1-X3 = Factor1
Sample Size: 200
Save Sigma in File MODEL1C
End of problem
Program Two Validation Sample and Compute CVI
Observed Variables: X1 X2 X3
Covariance Matrix
5.74
3.47 4.36
45.65 22.58 611.63
Sample Size: 200
Crossvalidate File MODEL1C
End of problem

Y102005.indb 232

3/22/10 3:26:15 PM

233

Model Validation

Factor

Figure 12.2
Single Factor Model (cross validation).

A single factor measurement model with three indicator variables is being
tested to see if it cross-validates using a randomly split sample of data
(Figure 12.2).
The low CVI value indicated that the measurement model holds for both
subsamples. The reduced computer output from the CVI cross validation
program is:
PROGRAM ONE CALIBRATION SAMPLE
COVARIANCE MATRIX TO BE ANALYZED
X1
X2
X3
X1
5.86
X2
3.12
3.32
X3
35.28 23.85 622.09
SI was written to file MODEL1C
PROGRAM TWO VALIDATION SAMPLE AND COMPUTE CVI
COVARIANCE MATRIX TO BE ANALYZED
X1
X2
X3
X1
5.74
X2
3.47
4.36
X3
45.65 22.58 611.63
MATRIX SIGMA
X1
X2
X3
X1
5.86
X2
3.12
3.32
X3
35.28 23.85 622.09
CROSS‑VALIDATION INDEX (CVI) = 0.38

The ECVI and CVI are most useful after a theoretically implied model has
an acceptable model fit, that is, when a specified model yields model-fit

Y102005.indb 233

3/22/10 3:26:16 PM

234

A Beginner’s Guide to Structural Equation Modeling

indices and parameter estimates that are meaningful with sufficient sample
size and power. The number of parameters, model complexity, and sample
size affect these cross validation indices; therefore, you should not routinely
discard other modeling considerations when you select the smaller ECVI of
two competing models, report the CVI from two subsamples, or report the
CVI across samples taken from a population. Currently, LISREL–SIMPLIS
computes ECVI for single sample expected cross validation, however, only
LISREL–SIMPLIS computes CVI for split sample cross validation.

12.3 Bootstrap
The bootstrap method treats a random sample of data as a substitute
for the population (pseudo population) and re-samples from it a specified number of times, to generate sample bootstrap estimates and standard errors. These sample bootstrap estimates and standard errors are
averaged and used to obtain a confidence interval around the average
of the bootstrap estimates. This average is termed a bootstrap estimator.
The bootstrap estimator and associated confidence interval are used to
determine how stable or good the sample statistic is as an estimate of the
population parameter. Obviously, if the random sample initially drawn
from the population is not representative, then the sample statistic and
corresponding bootstrap estimator obtained from re-sampling will yield
misleading results. The bootstrap approach is used in research when replication with additional sample data and/or cross validation with a split
sample is not possible. Fan (2003) demonstrates how the bootstrap method
is implemented in various software packages and its utility in correlation, regression, analysis of variance, and reliability. We present examples
using PRELIS.
12.3.1 PRELIS Graphical User Interface
Bootstrapping can be accomplished in two different ways using PRELIS
(Jöreskog & Sörbom, 1993; 1996b); LISREL–SIMPLIS program does not
provide bootstrap capabilities. Our first example will demonstrate the
use of the PRELIS graphical user interface. The second example will use
the PRELIS command language syntax (Jöreskog & Sörbom, 1996b, pp.
185–190). In our first bootstrap example, we select File, then Import Data
to import the SPSS saved file, dataex7.sav, located in the SPSSEX subfolder
in LISREL 8 Student Examples directory and save the PRELIS SYSTEM
FILE, dataex7.psf.

Y102005.indb 234

3/22/10 3:26:16 PM

Model Validation

Y102005.indb 235

235

3/22/10 3:26:17 PM

236

A Beginner’s Guide to Structural Equation Modeling

We now see the PRELIS SPREADSHEET with the raw data and the
PRELIS tool bar menu with several options from which to choose. We
select Statistics from the toolbar menu, and then Bootstrapping.

Once we select Bootstrapping, a dialog appears that permits us to specify the number of bootstrap samples, bootstrap fraction, and names for
saving the bootstrap matrix, means and standard deviations. The Syntax
button will create a PRELIS program that you can edit and save. The
Output Options button provides other formats for saving the data.

Y102005.indb 236

3/22/10 3:26:17 PM

237

Model Validation

The output provides us with the PRELIS command language syntax
program and descriptive statistics, as follows.
!PRELIS SYNTAX: Can be edited
SY=’C:\lisrel854\spssex\dataex7.PSF’
OU MA=KM SM=data7.cor ME=data7.me SD=data7.sd XM BS=100 SF=50
BM=data7.cor ME=data7.me SD=data7.sd
Bootstrap Correlation Matrix
NOSAY
VOTING
COMPLEX
NOCARE
TOUCH
INTEREST

NOSAY
1.000
0.292
0.259
0.462
0.386
0.408

VOTING

COMPLEX

NOCARE

TOUCH INTEREST

1.000
0.276
0.263
0.180
0.239

1.000
0.442
0.294
0.368

1.000
0.669
0.710

1.000
0.640

1.000

Descriptive Statistics
Variable
NOSAY
VOTING
COMPLEX
NOCARE
TOUCH
INTEREST

Mean
0.000
0.000
0.000
0.000
0.000
0.000

St. Dev.
1.000
1.000
1.000
1.000
1.000
1.000

12.3.2 LISREL and PRELIS Program Syntax
In our second example, we use LISREL and PRELIS command language
syntax in various programs to further elaborate the bootstrap method. We
first run a LISREL program using the original sample data. The raw-data
file, efficacy.raw, is provided with LISREL and used in other examples in
the PRELIS2 User’s Reference Guide (Jöreskog & Sörbom, 1996b). A twofactor model is specified with six factor loadings estimated; three for each
of the factors (see the MO and FR command lines). The LISREL program
is written as:
Estimate factor loadings for model from file efficacy.raw
DA NI=6 NO=297 ME=GLS
RA=efficacy.raw FO;(6F1.0)
CO ALL
MO NX=6 NK=2
FR LX(1,1) LX(2,1) LX(3,1) LX(4,2) LX(5,2) LX(6,2)
OU MA=CM

Y102005.indb 237

3/22/10 3:26:17 PM

238

A Beginner’s Guide to Structural Equation Modeling

The variance-covariance matrix to be analyzed is indicated as:
VAR
VAR
VAR
VAR
VAR
VAR

1
2
3
4
5
6

0.60
0.16
0.11
0.23
0.16
0.19

0.59
0.14
0.14
0.08
0.11

0.59
0.21 0.57
0.14 0.30 0.49
0.17 0.34 0.27 0.53

The six factor loadings for the two factor model specified are estimated as:

LX(1,1)
LX(2,1)
LX(3,1)
LX(4,2)
LX(5,2)
LX(6,2)

Estimate

Standard Error

0.43
0.30
0.37
0.63
0.48
0.55

0.05
0.05
0.05
0.04
0.04
0.04

Then, to compute bootstrap estimates of the factor loadings for the twofactor model with three indicators per factor, the raw data file is read into a
PRELIS program with the number of variables, number of cases, and estimation method specified (DA NI=6 NO=297, ME=GLS). In this example,
the PRELIS program reads in a raw data file containing 6 variables and
297 cases with the generalized least-squares estimation method selected
[RA = efficacy.raw FO;(6F1.0)]. The PRELIS program then generates 10 covariance matrices using the generalized least-squares estimation method.
The number of bootstrap samples to be taken is specified (BS=10), and
these samples are randomly drawn from the raw data file with replacement. A 100% resampling (SF=100) of the raw data file is specified. The
10 covariance matrices are output into a bootstrap save file (BM = efficacy.
cm) for further analysis by another LISREL program. This output file is in
ASCII format and can be examined. The PRELIS program is:
Generate 10 covariance matrices from file efficacy.raw
DA NI=6 NO=297 ME=GLS
RA=efficacy.raw FO;(6F2.0)
OU MA=CM BS=10 SF=100 BM=efficacy.cm

The first two variance-covariance matrices output into the file efficacy.cm are:
VAR
VAR
VAR
VAR

Y102005.indb 238

1
2
3
4

1.00
0.27 1.00
0.26 0.26 1.00
0.46 0.25 0.42 1.00

3/22/10 3:26:17 PM

239

Model Validation

VAR 5 0.38 0.16 0.27 0.64 1.00
VAR 6 0.43 0.26 0.36 0.72 0.63 1.00
VAR
VAR
VAR
VAR
VAR
VAR

1
2
3
4
5
6

1.00
0.32
0.11
0.40
0.35
0.34

1.00
0.22
0.26
0.18
0.22

1.00
0.45 1.00
0.36 0.68 1.00
0.32 0.72 0.68 1.00

Notice that the diagonal values indicate variances equal to 1.0, whereas
the off-diagonal values indicate the covariance terms. The manipulation
of raw data (recoding variables, selecting cases, transformations) and the
treatment of missing data (imputation method and/or deleting cases listwise) should be specified and handled in this program prior to bootstrap
estimation. The researcher can also specify the type of matrix and estimation method desired in the PRELIS program.
The saved file, efficacy.cm, is next read by a LISREL program (CM = efficacy.cm) to estimate 10 sets of six factor loadings for the two-factor model.
The output from this program indicates the 10 different bootstrap sampled
covariance matrices read from the file, as well as parameter estimates, fit
indices, and so forth. (Note: The output is no different from running 10
separate stacked programs.)
The LISREL program is written as:
Estimate 10 sets of 6 factor loadings for two factor model
DA NI=6 NO=297 RP=10
CM=efficacy.cm
MO NX=6 NK=2
FR LX(1,1) LX(2,1) LX(3,1) LX(4,2) LX(5,2) LX(6,2)
OU LX=efficacy.lx

The LISREL program indicates that 6 variables and 297 cases were used
to compute the 10 covariance matrices that are read in from the saved file
(CM = efficacy.cm). The program is run 10 times (RP = 10), once for each covariance matrix saved in the file. The model specifies six variables and two
factors (MO NX=6 NK=2). The parameters (factor loadings) to be estimated
indicate that the first three variables define one factor and the last three
variables define a second factor (see the FR command line, which indicates
elements in the matrix to be free or estimated). The 10 sets of six factor loadings are computed and output in a saved file (OU LX=efficacy.lx).
The saved file is then read by the following PRELIS program to generate
the bootstrap estimates and standard errors for the six factor loadings in
the model:
Analyze 10 sets of 6 factor loadings from file efficacy.lx
DA NI=12

Y102005.indb 239

3/22/10 3:26:18 PM

240

A Beginner’s Guide to Structural Equation Modeling

LA
‘LX(1,1)’ ‘LX(1,2)’ ‘LX(2,1)’ ‘LX(2,2)’ ‘LX(3,1)’ ‘LX(3,2)’
‘LX(4,1)’ ‘LX(4,2)’ ‘LX(5,1)’ ‘LX(5,2)’ ‘LX(6,1)’ ‘LX(6,2)’
RA=efficacy.lx
SD ‘LX(1,2)’ ‘LX(2,2)’ ‘LX(3,2)’ ‘LX(4,1)’ ‘LX(5,1)’ ‘LX(6,1)’
CO ALL
OU MA=CM

The PRELIS program analyzes the 10 sets of six factor-loading bootstrap estimates and outputs summary statistics. Notice that we used the SD command
to delete the other six factor loadings that were set to zero in the two-factor
model. For our example, the bootstrap estimator and standard deviation for
the six factor loadings (three-factor loadings for each factor) were:
UNIVARIATE SUMMARY STATISTICS FOR CONTINUOUS VARIABLES
VARIABLE
MEAN S. D.
LX(1,1)
0.298 0.322
LX(2,1)
0.447 0.459
LX(3,1)
0.207 0.230
LX(4,2)
0.373 0.384
LX(5,2)
0.251 0.260
LX(6,2)
0.403 0.415

These values can be used to form confidence intervals around the original sample factor-loading estimates to indicate how stable or good the
estimates are as estimates of population values. Rather than further discuss the PRELIS and LISREL command language syntax program setups for bootstrapping, we refer you to the manual and excellent help
examples in the software for various straightforward data set examples
and output explanations. These two examples were intended only to
provide a basic presentation of the bootstrap method in structural equation modeling. Lunneborg (1987) provided additional software to compute bootstrap estimates for means, correlations (bivariate, multivariate,
part, and partial), regression weights, and analysis-of-variance designs,
to name a few. Stine (1990) provided a basic introduction to bootstrapping methods, and Bollen and Stine (1993) gave a more in-depth discussion of bootstrap in structural equation modeling. Mooney and Duval
(1993) also provided an overview of bootstrapping methods, gave a basic
algorithm and program for bootstrapping, and indicated other statistical packages that have bootstrap routines. We therefore refer you to
these references, as well as others presented in this section, for a better coverage of the background, rationale, and appropriateness of using
bootstrap techniques.

Y102005.indb 240

3/22/10 3:26:18 PM

Model Validation

241

12.4 Summary
In this chapter, our concern was model validation. A theoretical model
requires validation on additional random samples of data. We refer to
this as replication and demonstrated how multiple samples could be tested
against the specified theoretical model. In the absence of replication, cross
validation and bootstrap techniques were discussed as a means of validating a theoretical model.
The chapter began with a look at replication involving the testing of
the multiple samples of data against the theoretical model, followed by
single sample (ECVI) and split-sample (CVI) cross-validation techniques.
We also introduced the bootstrap method to assess the stability of our
parameter estimates and standard errors, especially given different distributional assumptions.
We hope that our discussion of these model validation topics in structural equation modeling has provided you with a basic overview and
introduction to these methods. We encourage you to read the references
provided at the end of the chapter and run some of the program setups
provided in the chapter. We further hope that the basic introduction in
this chapter will permit you to read the research literature and better
understand the topics presented in the chapter. We now turn our attention to chapters 13 to 16 where we present various advanced SEM applications to demonstrate the variety of research designs and research
questions that can be addressed using structural equation modeling.
Exercises
1. Test whether the following three variance-covariance matrices
fit the theoretical confirmatory factor model in Figure 12.3 using
LISREL–SIMPLIS. The sample size is 80 for each sample. The
variables are entered in order as: SOFED (father’s education),
SOMED (mothers’ education), SOFOC (father’s occupation),
FAFED (father’s education), MOMED (mother’s education), and
FAFOC (father’s occupation).
Sample 1
5.86
3.12 3.32
35.28 23.85 622.09
4.02 2.14 29.42 5.33
2.99 2.55 19.20 3.17 4.64
35.30 26.91 465.62 31.22 23.38 546.01
Sample 2
8.20
3.47 4.36

Y102005.indb 241

3/22/10 3:26:18 PM

242

A Beginner’s Guide to Structural Equation Modeling

45.65 22.58 611.63
6.39 3.16 44.62 7.32
3.22 3.77 23.47 3.33 4.02
45.58 22.01 548.00 40.99 21.43 585.14
Sample 3
5.74
1.35 2.49
39.24 12.73 535.30
4.94 1.65 37.36 5.39
1.67 2.32 15.71 1.85 3.06
40.11 12.94 496.86 38.09 14.91 538.76
a. Run individual program for sample 1 to determine CFA model
and report CFA parameters.
b. Run individual programs with CFA model on sample 2 and
sample 3 and report CFA parameters.
c. Run multiple-sample program to test factor loadings and factor
correlations invariant (equal) with unequal error variances and
report individual and global chi-square values.
d. Interpret your results.
2. For Exercise #1, Report the single sample expected cross validation index (ECVI). Given a sample size of 80, would you split the
sample and cross validate the model using CVI?

SOFED

err_sf

FAFED

err_ff

SOMED

err_sm

MOMED

err_mf

SOFOC

err_sc

FAFOC

err_fc

FED

MED

FOC

Figure 12.3
Multiple Sample Confirmatory Factor Model.

Y102005.indb 242

3/22/10 3:26:18 PM

Model Validation

243

References
Arbuckle, J. L., & Wothke, W. (1999). AMOS 4.0 User’s Guide. Chicago, IL:
Smallwaters Corporation.
Bandalos, D. (1993). Factors influencing the cross-validation of confirmatory factor
analysis models. Multivariate Behavioral Research, 28, 351–374.
Benson, J., & Bandalos, D. (1992). Second-order confirmatory factor analysis of the
reactions to tests’ scale with cross-validation. Multivariate Behavioral Research,
27, 459–487.
Benson, J, Moulin-Julian, M., Schwarzer, C., Seipp, B., & El-Zahhar, N. (1992).
Cross-validation of a revised test anxiety scale using multi-national samples.
In K. Hagtvet (Ed.), Advances in test anxiety research: Vol. 7 (pp. 62–83). Lisse,
Netherlands: Swets & Zeitlinger.
Benson, J., & El-Zahhar, N. (1994). Further refinement and validation of the revised
test anxiety scale. Structural Equation Modeling: A Multidisciplinary Journal,
1(3), 203–221.
Bentler, P. M., & Wu, E. (2002). EQS for Windows User’s Guide. Encino, CA:
Multivariate Software.
Bollen, K. A. (1989). Structural equations with latent variables. New York: Wiley.
Bollen, K. A., & Stine, R. A. (1993). Bootstrapping goodness-of-fit measures in
structural equation models. In K. A. Bollen, & J. S. Long (Eds.), Testing structural equation models (pp. 66–110). Newbury Park, CA: Sage.
Browne, M., & Cudeck, R. (1989). Single sample cross-validation indices for covariance structures. Multivariate Behavioral Research, 24, 445–455.
Browne, M., & Cudeck, R. (1993). Alternative ways of assessing model fit. In K. A. Bollen
& J. S. Long (Eds.), Testing structural equation models. Newbury Park, CA: Sage.
Cole, D. A., & Maxwell, S. E. (1985). Multitrait‑multimethod comparisons across
populations: A confirmatory factor analytic approach. Multivariate Behavioral
Research, 20, 389–417.
Cudeck, R., & Browne, M. W. (1983). Cross-validation of covariance structures.
Multivariate Behavioral Research, 18, 147–167.
Fan, X. (2003). Using commonly available software for bootstrapping in both substantive and measurement analysis. Educational and Psychological Measurement,
63(1), 24–50.
Faulbaum, F. (1987). Intergroup comparisons of latent means across waves.
Sociological Methods and Research, 15, 317–335.
Hayduk, L. A. (1987). Structural equation modeling with LISREL: Essentials and
advances. Baltimore: Johns Hopkins University Press.
Jöreskog, K. G., & Sörbom, D. (1993). Bootstrapping and Monte Carlo experimenting
with PRELIS2 and LISREL8. Chicago: Scientific Software International.
Jöreskog, K. G., & Sörbom, D. (1996a). LISREL8 user’s reference guide. Chicago:
Scientific Software International.
Jöreskog, K. G., & Sörbom, D. (1996b). PRELIS2: User’s reference guide. Chicago:
Scientific Software International.
Jöreskog, K. G., & Sörbom, D. (1996c). LISREL8: Structural equation modeling with
the SIMPLIS command language. Hillsdale, NJ: Lawrence Erlbaum.

Y102005.indb 243

3/22/10 3:26:18 PM

244

A Beginner’s Guide to Structural Equation Modeling

Lomax, R. G. (1983). A guide to multiple sample equation modeling. Behavior
Research Methods and Instrumentation, 15, 580–584.
Lomax, R. G. (1985). A structural model of public and private schools. Journal of
Experimental Education, 53, 216–236.
Lunneborg, C. E. (1987). Bootstrap applications for the behavioral sciences: Vol. 1.
Psychology Department, University of Washington, Seattle.
McArdle, J. J., & Epstein, D. (1987). Latent growth curves within developmental
structural equation models. Child Development, 58, 110–133.
Mooney, C. Z., & Duval, R. D. (1993). Bootstrapping: A nonparametric approach to
statistical inference. Sage University Series on Quantitative Applications in the
Social Sciences, 07-097. Beverly Hills, CA: Sage.
Stine, R. (1990). An introduction to bootstrap methods: Examples and ideas. In J.
Fox. & J. S. Long (Eds.), Modern methods of data analysis (pp. 325–373). Beverly
Hills, CA: SAGE.

Y102005.indb 244

3/22/10 3:26:19 PM

13
Multiple Sample, Multiple Group,
and Structured Means Models

Key Concepts
Testing for parameter differences between samples of data
Testing parameter differences between groups
Testing hierarchical intercept and slope differences in nested groups
In previous chapters, we have learned about the basics of structural equation modeling. In this chapter and subsequent chapters, we will consider
other SEM models that demonstrate the variety of applications suitable
for structural equation modeling. You should be aware, however, that
our discussion will only introduce these SEM models. You are encouraged to explore other examples and applications reported in books
(Marcoulides & Schumacker, 1996; Marcoulides & Schumacker, 2001),
LISREL software examples, and the references at the end of this chapter.
Our intention is to provide a basic understanding of the applications in
this chapter to further your interest in the structural equation modeling
approach. We have used LISREL–SIMPLIS program examples to better
illustrate each application.

13.1 Multiple Sample Models
The multiple samples approach was explained in a previous chapter, but
related to testing measurement invariance in a measurement model. We
expand on the multiple sample approach here to include testing a model
for differences in parameter estimates across samples of data. The theoretical model is in Figure 13.1a.
The data set we used for our multiple sample approach can be found
in SPSS 16 Sample folder: C:\Program Files\SPSSInc\SPSS16\Samples\
245

Y102005.indb 245

3/22/10 3:26:19 PM

246

A Beginner’s Guide to Structural Equation Modeling

weight

mpg

power

Figure 13.1a
Multiple Samples Model.

Cars.sav. The Cars.sav data set estimates miles per gallon (mpg) based on
various vehicle characteristics (weight, horsepower, engine displacement,
year of vehicle, etc.). For our purposes we selected mile per gallon as the
dependent variable with vehicle weight and horsepower as independent
predictor variables. The original data set contains N = 406; however, only
N = 392 are useable because of 14 missing cases (8 due to dependent variable missingness and 6 due to independent variable missingness).
The descriptive statistics for the dependent and independent variables
are shown in Table 13.1. The average was 23.45 miles per gallon with an
approximate +/−7.8 miles per gallon standard deviation.
Table 13.2 reports the multiple regression prediction results that yielded
an R2 = .675 (F = 404.583; df = 2, 389; p = .0001). Our interpretation would suggest that two-thirds of the miles per gallon variation can be explained by a
vehicles weight and horsepower. The negative beta coefficients are expected
because as weight and horsepower increase, miles per gallon decrease.
Our interest in multiple samples is to compare the parameter estimates
of each sample to determine whether they differ significantly. We therefore took two random samples without replacement from the Cars.sav

Table 13.1
Complete Sample Descriptive Statistics (N = 392)
Variable
Miles per gallon
Vehicle weight
Horse power
Mean
SD

Y102005.indb 246

Miles Per Gallon

Vehicle Weight

1.0
–.807
–.771

1.0
.857

23.45
7.805

2967.38
852.294

Horsepower

1.0
104.21
38.233

3/22/10 3:26:19 PM

247

Multiple Sample, Multiple Group, and Structured Means Models

Table 13.2
Multiple Regression Complete Sample
Unstandardized
Coefficients

Constant
Vehicle Weight
Horsepower

Standardized
Coefficients

95% Confidence
Interval for B

Std
Error

Lower
CI

Upper
CI

44.777
−.005
−.061

.825
.001
.011

−.551
−.299

54.307
−9.818
−5.335

.0001
.0001
.0001

43.156
−.006
−.084

46.398
−.004
−.039

data.* The descriptive statistics for both samples are indicated below in
Table 13.3 and Table 13.4, respectively.
Sample 1
Table 13.3
Sample 1 Descriptive Statistics (N = 206)
Variable
Miles per gallon
Vehicle weight
Horse power
Mean
SD

Miles Per Gallon

Vehicle Weight

1.0
−.821
−.778

1.0
.865

23.94
8.140

2921.67
835.421

Horsepower

1.0
104.23
41.129

Sample 2
Table 13.4
Sample 2 Descriptive Statistics (N = 188)
Variable
Miles per gallon
Vehicle weight
Horse power
Mean
SD

Miles Per Gallon

Vehicle Weight

Horsepower

1.0
−.823
−.760

1.0
.855

1.0

23.59
7.395

2952.02
805.372

102.72
36.234

The SPSS multiple regression analyses are in Table 13.5 and Table 13.6,
respectively, for the two samples of data. We see from the SPSS multiple
*

See Chapter Footnote for SPSS details on selecting random samples from Cars.sav.

Y102005.indb 247

3/22/10 3:26:20 PM

248

A Beginner’s Guide to Structural Equation Modeling

Table 13.5
Sample 1 Multiple Regression Results (N = 206)
Unstandardized Standardized
Coefficients
Coefficients

Constant
Vehicle Weight
Horsepower

95% Confidence
Interval for B

Std
Error

Lower
CI

Upper
CI

46.214
−.006
−.054

1.193
.001
.015

−.585
−.272

38.723
−7.550
−3.509

.0001
.0001
.0001

43.861
−.007
−.084

48.568
−.004
−.024

R2 = .692 (F = 228.206; df = 2, 203; p = .001)

regression analysis of the complete data (N = 292) what our sample results
provide in terms of R2 values, F value, and regression coefficients. We also
can visually compare our two individual sample SPSS regression analyses. The results appear to be very similar. Structural equation modeling
software, however, provides the capability of testing whether our results
(parameter estimates) are statistically different.
LISREL provides the ability to compare both samples rather than having to run separate multiple regression programs on each sample and
hand calculate a t-test or z-test for differences in the regression weights.
The LISREL multiple sample approach is therefore presented to show how
to stack or include each program with different samples of data.
Table 13.6
Sample 2 Multiple Regression Results (N = 188)
Unstandardized
Coefficients

Constant
Vehicle weight
Horsepower

Standardized
Coefficients

95% Confidence
Interval for B

Std
Error

45.412
−.006
−.043

1.166
.001
.016

−.642
−.212

38.957
−8.114
−2.675

.0001
.0001
.0001

Lower Upper
CI
CI
43.112
−.007
−.075

47.712
−.004
−.011

R2 = .689 (F = 204.502; df = 2, 185; p = .001).

Regression model comparing two samples
Group 1: Sample 1
Observed variables: mpg weight power

Y102005.indb 248

3/22/10 3:26:20 PM

Multiple Sample, Multiple Group, and Structured Means Models

249

Sample Size: 206
Correlation Matrix
1.0
−.821 1.0
−.778 .865 1.0
Means 23.94 2921.67 104.23
Standard Deviations 8.140 835.421 41.129
Equations:
mpg = weight power
Group 2: Sample 2
Observed variables: mpg weight power
Sample Size: 188
Correlation Matrix
1.0
−.823 1.0
−.760 .855 1.0
Means: 23.59 2952.02 102.72
Standard Deviations: 7.395 805.372 36.234
Path Diagram
End of Problem
The LISREL multisample output in Figure 13.1b reveals that the chisquare test is nonsignificant (c2 = 2.01, df = 3, p = .57), which indicates that
the two samples do not have statistically different parameter estimates
in the regression model. Another way of thinking about these results is

1.03

weight
–0.61

0.93

mpg

0.31

–0.25
1.12

power
Chi-Square = 2.01, df =3, p = .57

Figure 13.1b
Multiple Samples Output.

Y102005.indb 249

3/22/10 3:26:20 PM

250

A Beginner’s Guide to Structural Equation Modeling

that both samples fit the theoretical model equally. The regression coefficient in common for weight predicting mpg was −.61; the individual
regression weights in SPSS were estimated as −.585 and −.642, respectively. So, it seems reasonable to have a common regression beta weight
of –.61. Looking at the regression weight for power predicting mpg, we
find a common regression coefficient of −.25. The individual regression
weights in SPSS were estimated as −.272 and −.212, respectively. So, it
seems reasonable to have a common regression beta weight of –.25. Also,
notice that the error of prediction for mpg is .31 (1 – R 2), which means that
the common model R2 = .69. We find that for each individual sample, the
R2 values were .692 and .689, respectively. So, once again, the common R2
value of .69 is reasonable.
The SEM modeling approach is useful for testing whether samples of
data yield similar or different parameter estimates, whether comparing
multiple regression equations, path models, confirmatory factor models,
or structural equation models.

13.2 Multiple Group Models
Multiple group models are set up the same way as multiple sample models. You are basically applying a single specified model to either one or
more samples of data or in the case of multiple groups, one or more
groups. This type of SEM modeling permits testing for group differences
in the specified model or testing for differences in specific parameter estimates by imposing constraints. For example, Lomax (1985) examined a
model for schooling using the High School and Beyond (HSB) database.
The model included home background, academic orientation, extracurricular activity, achievement, and educational and occupational aspirations as latent variables. The research determined the extent to which
the measurement and structural equation models fit both a sample of
public school students and a sample of private school students and also
examined whether model differences existed between the two groups.
The multiple group model analysis should first establish the acceptance
of the measurement models and measurement invariance for the groups
before hypothesizing any statistically significant difference in coefficients between groups. A LISREL–SIMPLIS multiple group example is
presented based on an example in Arbuckle and Wothke (2003). The multiple group model is specified to examine the perceived attractiveness

Y102005.indb 250

3/22/10 3:26:21 PM

Multiple Sample, Multiple Group, and Structured Means Models

251

and perceived academic ability differences between a sample of 209 girls
and 207 boys.
The LISREL–SIMPLIS program is constructed to include the GROUP
command to distinguish between the two groups of data. The observed
variables, sample size, means, standard deviations, and correlation
matrix are given for each group. The LISREL–SIMPLIS program provides a test of a common model when you only include the EQUATIONS
or RELATIONSHIP command in the first group. The computer output
yields a common model with the parameter estimates. If you wish to
have separate models, hence separate parameter estimates for each group,
you would run each program separately with the same EQUATIONS or
RELATIONSHIP command in both programs.
13.2.1 Separate Group Models
We will begin by first running a LISREL–SIMPLIS program that provides
separate path analysis estimates for girls and boys. The LISREL–SIMPLIS
program would be run as follows:
Multiple Group Path Model Analysis
Group 1: Girls
Observed Variables academic attract gpa height weight rating
Sample Size = 209
Means .12 .42 10.34 .00 94.13 2.65
Standard Deviation .16 .49 3.49 2.91 19.32 1.01
Correlation Matrix
1.00
.50 1.00
.49 .32 1.00
.10 −.03 .18 1.00
.04 −.16 -.10 .34 1.00
.09 .43 .15 −.16 −.27 1.00
Equation:
academic = gpa attract
attract = academic height weight rating
Let the errors of academic and attract correlate
Group 2: Boys
Observed Variables academic attract gpa height weight rating
Sample Size = 207
Means: .10 .44 8.63 .00 101.91 2.59
Standard Deviations: .16 .49 4.04 3.41 24.32 .97
Correlation Matrix
1.00

Y102005.indb 251

3/22/10 3:26:21 PM

252

A Beginner’s Guide to Structural Equation Modeling

.49 1.00
.58 .30 1.00
−.02 .04 −.11 1.00
−.11 -.19 -.16 .51 1.00
.11 .28 .13 .06 −.18 1.00
Equation:
academic = gpa attract
attract = academic height weight rating
Let the errors of academic and attract correlate
Number of Decimals = 3
Path diagram
End of problem

Computer Output
The annotated computer output for girls and boys multiple-group model
(Figure 13.2a and Figure 13.2b) results are listed below:
GIRLS
Structural Equations
academic = 0.0257* attract + 0.0212*gpa, Errorvar.= 0.0175, R² = 0.296
(0.0427)
(0.00329)
(0.00213)
0.603
6.440
8.196
attract = 1.688*academic − 0.000248*height – 0.00169*weight + 0.175*rating,
(0.362)
(0.0102)
(0.00154)
(0.0287)
4.666
−0.0244
−1.097
6.085
Errorvar.= 0.155 , R² = 0.386
(0.0110)
14.044
Error Covariance for attract and academic = −0.010
(0.00979)
−0.982
Group Goodness-of-Fit Statistics
Contribution to Chi-Square = 3.773
Percentage Contribution to Chi-Square = 66.580
Root Mean Square Residual (RMR) = 0.105
Standardized RMR = 0.0276
Goodness-of-Fit Index (GFI) = 0.994

Y102005.indb 252

3/22/10 3:26:21 PM

253

Multiple Sample, Multiple Group, and Structured Means Models

0.86

gpa
0.50

–0.15

academic
–0.08
–0.14

0.84

height

–0.27
–0.15

0.77

–0.00
weight

–0.24
1.04

0.68

–0.08

0.55

0.08

attract

–0.12

0.64

0.35
rating

Figure 13.2a
Multiple Group Model: girls.

BOYS
Structural Equations
academic = 0.00657*attract + 0.0227*gpa, Errorvar.= 0.0175, R² = 0.338
(0.0481)
(0.00288)
0.00213)
0.137
7.882
8.196
attract = 1.381*academic + 0.0179*height – 0.00341*weight + 0.0975*rating,
(0.303)
(0.00955)
(0.00136)
(0.0295)
4.560
1.875
−2.504
3.301
Errorvar.= 0.155 , R² = 0.323
(0.0110)
14.044
Error Covariance for attract and academic = −0.001
(0.00989)
−0.095

Y102005.indb 253

3/22/10 3:26:21 PM

254

A Beginner’s Guide to Structural Equation Modeling

Group Goodness-of-Fit Statistics
Contribution to Chi-Square = 1.894
Percentage Contribution to Chi-Square = 33.420
Root Mean Square Residual (RMR) = 0.0223
Standardized RMR = 0.0183
Goodness-of-Fit Index (GFI) = 0.997

1.15

gpa
0.54

–0.13

academic
–0.19
0.14

1.16

height

0.61
0.06

1.23

0.12
weight

–0.20
0.96

0.68

–0.15

0.45

0.02

attract

–0.01

0.64

0.20
rating

Figure 13.2b
Multiple Group Model: boys.

The results indicate different parameter estimates for the girl’s data and
the boy’s data when applied to the model. For example, the reciprocal path
coefficients between academic and attract for the girl’s data were p12 = .55
and p21 = .08, whereas for the boy’s data these same path coefficients were
.45 and .02, respectively. The girl’s data fit the path model as indicated by
the nonsignificant chi-square value (c2 = 3.773), and the boy’s data also fit
the path model as indicated by their nonsignificant chi-square value (c2 =
1.894). The Global Fit Statistics indicated a chi-square for the hypothesis of
unequal (separate) parameter estimates in the path model (c2 = 5.667, df =
6, p = .461). You will notice that the separate chi-square values for the girls’
and boys’ path model results will add up to this global chi-square value:
c2 = 3.773 (girls) + 1.894 (boys) = 5.667. Our primary interest, however, is in
testing a hypothesis about whether the groups have equal (same) parameter estimates in the path model.

Y102005.indb 254

3/22/10 3:26:22 PM

Multiple Sample, Multiple Group, and Structured Means Models

255

13.2.2 Similar Group Model
LISREL–SIMPLIS uses the GROUP command (GROUP must be followed
by a number) and does not use the EQUATION commands in the second
group, when testing whether the two groups share a common path model.
The LISREL–SIMPLIS program would now be as follows:
Multiple Group Path Model Analysis
Group 1: Girls
Observed Variables academic attract gpa height weight rating
Sample Size = 209
Means .12 .42 10.34 .00 94.13 2.65
Standard Deviation .16 .49 3.49 2.91 19.32 1.01
Correlation Matrix
1.00
.50 1.00
.49 .32 1.00
.10 −.03 .18 1.00
.04 −.16 −.10 .34 1.00
.09 .43 .15 −.16 −.27 1.00
Equation:
academic = gpa attract
attract = academic height weight rating
Let the errors of academic and attract correlate
Group 2: Boys
Observed Variables academic attract gpa height weight rating
Sample Size = 207
Means: .10 .44 8.63 .00 101.91 2.59
Standard Deviations: .16 .49 4.04 3.41 24.32 .97
Correlation Matrix
1.00
.49 1.00
.58 .30 1.00
−.02 .04 −.11 1.00
−.11 -.19 −.16 .51 1.00
.11 .28 .13 .06 −.18 1.00
Number of Decimals = 3
Path diagram
End of problem

Computer Output
Structural Equations
academic = 0.0167*attract + 0.0221*gpa, Errorvar. = 0.0174, R² = 0.290
(0.0404)
(0.00237)
(0.00217)
0.414
9.330
8.039

Y102005.indb 255

3/22/10 3:26:22 PM

256

A Beginner’s Guide to Structural Equation Modeling

attract = 1.439*academic + 0.00863*height – 0.00256*weight + 0.142*rating,
(0.233)
(0.00687)
(0.00102)
(0.0204)
6.189
1.255
−2.499
6.985
Errorvar. = 0.156 , R² = 0.346
(0.0109)
14.309
Error Covariance for attract and academic = −0.003
(0.00796)
−0.429
GIRLS
Group Goodness-of-Fit Statistics
Contribution to Chi-Square = 6.739
Percentage Contribution to Chi-Square = 57.949
Root Mean Square Residual (RMR) = 0.0920
Standardized RMR = 0.0320
Goodness-of-Fit Index (GFI) = 0.989
BOYS
Group Goodness-of-Fit Statistics
Contribution to Chi-Square = 4.890
Percentage Contribution to Chi-Square = 42.051
Root Mean Square Residual (RMR) = 0.0276
Standardized RMR = 0.0249
Goodness-of-Fit Index (GFI) = 0.992
Global Goodness-of-Fit Statistics
Degrees of Freedom = 13
Minimum Fit Function Chi-Square = 11.629 (P = 0.558)
Normal Theory Weighted Least Squares Chi-Square = 11.699
(P = 0.552)
Root Mean Square Error of Approximation (RMSEA) = 0.0
90 Percent Confidence Interval for RMSEA = (0.0 ; 0.0633)
P-Value for Test of Close Fit (RMSEA < 0.05) = 0.876
Normed Fit Index (NFI) = 0.975
When the path diagram window is open you will see a window labeled,
Groups: Multiple Group Path Model. The first path model is for GIRLS. All
of the parameters specified in the EQUATIONS command are set equal
between the two groups. Only the covariance among the observed variables is free to vary.

Y102005.indb 256

3/22/10 3:26:22 PM

Multiple Sample, Multiple Group, and Structured Means Models

257

When you scroll down to select Group 2: Boys in the Groups window, you
will then see the path diagram for the boys. You will see that the parameter estimates are equal for all the paths specified in the EQUATIONS
command. The only parameters free to vary (be different) are the covariance among the observed variables.

You will notice that the parameter estimates are the same in both groups.
For example, p12 = .47 and p21 = .05. The individual chi-square values for each
group also add up to the global chi-square statistic for this common model.

Y102005.indb 257

3/22/10 3:26:23 PM

258

A Beginner’s Guide to Structural Equation Modeling

The chi-square for the girls was c2 = 6.739 and the chi-square for the boys
was c2 = 4.890, which yield the global chi-square value of c2 = 11.629, df =
13, p = .558. These results indicated that both sets of data fit the path model
based on the hypothesis of similar path coefficients in the path model.
13.2.3 Chi-Square Difference Test
It is possible to compute a chi-square difference test between the two path
model analyses. Recall that the first LISREL–SIMPLIS program analysis
tested a hypothesis of unequal parameter estimates, while the second
LISREL–SIMPLIS program analysis tested a hypothesis of equal parameter estimates. You can compute a chi-square difference test between these
two models by using an EXCEL spreadsheet program, CV.XLS. You will
find this EXCEL program by going to the LISREL 8.8 Student Examples
folder on the C:/directory, and then finding the WORKSHOP folder.
Open the CV.XLS program outside the LISREL–SIMPLIS program, otherwise it will crash and cause an error message. All you have to do is enter
the Global Chi-Square value from the analysis of equal parameter estimates
(c2 = 11.629, df = 13) and the Global Chi-Square value from the analysis of
unequal parameter estimates (c2 = 5.667, df = 6). The program calculates
the difference in the chi-square values and associated p-values.
The chi-square difference was c2 = 5.962, df = 7, p = .544, which indicated
no difference between the two model analyses. This implies that the girls’
and boys’ data separately fit the path model, as well as both data sets fit
a common path model. A different path model analysis might examine
other variables besides gender that produce different results, for example,
Caucasian versus African-American path models.
The dialog box for the chi-square difference test should look like the
one below.

Y102005.indb 258

3/22/10 3:26:23 PM

Multiple Sample, Multiple Group, and Structured Means Models

259

13.3 Structured Means Models
The structured means model is yet another special type of SEM application that is used to test group mean differences in observed and/or latent
variables. Mean differences between observed variables in SEM is similar
to analysis of variance and covariance techniques. However, mean difference in latent variables is unique to SEM.
13.3.1 Model Specification and Identification
The structured means model example (Figure 13.3) is from LISREL–
SIMPLIS and uses the program EX13B.SPL (Jöreskog & Sörbom, 1993,
EX13B.SPL). The structured means model examines the mean difference
between academic and nonacademic boys in 5th and 7th grades on a latent
variable, verbal ability. The structured means model is diagrammed below
where writing and reading scores measure each latent variable at the 5th
grade (Writing5 and Reading5) and 7th grade (Writing7 and Reading7).
Two LISREL–SIMPLIS programs will need to be run to test the mean
difference between the latent variables, Verbal5 and Verbal7. The first program indicates the observed variables and equations that relate to the
structured means model diagram. The coefficient CONST is used to designate the means in the equations for the observed variables and the latent
variables, respectively. The first program also includes the sample size,
covariance matrix, and means for the first group (academic boys). The second program includes the sample size, covariance matrix, and means for
the second group (nonacademic boys). In addition, the second program

Writing7

err_w7

Reading7

err_r7

Writing5

err_w5

Reading5

err_r5

Verbal7

Verbal5

Figure 13.3
Structured Means Model.

Y102005.indb 259

3/22/10 3:26:24 PM

260

A Beginner’s Guide to Structural Equation Modeling

establishes a test of the mean differences between the latent variables in
the Relationship command:
Relationships:
Verbal5 = CONST
Verbal7 = CONST
The first and second LISREL–SIMPLIS programs are both stacked into
one complete program, but identified separately using the GROUP command, as follows:
Group ACADEMIC: Reading and Writing, Grades 5 and 7
Observed Variables: READING5 WRITING5 READING7 WRITING7
Covariance Matrix
281.349
184.219 182.821
216.739 171.699 283.289
198.376 153.201 208.837 246.069
Means 262.236 258.788 275.630 269.075
Sample Size: 373
Latent Variables: Verbal5 Verbal7
Relationships:
READING5 = CONST + 1*Verbal5
WRITING5 = CONST + Verbal5
READING7 = CONST + 1*Verbal7
WRITING7 = CONST + Verbal7
Group NONACADEMIC: Reading and Writing, Grades 5 and 7
Covariance Matrix
174.485
134.468 161.869
129.840 118.836 228.449
102.194 97.767 136.058 180.460
Means 248.675 246.896 258.546 253.349
Sample Size: 249
Relationships:
Verbal5 = CONST
Verbal7 = CONST
Set the Error Variances of READING5 - WRITING7 free
Set the Variances of Verbal5 - Verbal7 free
Set the Covariance between Verbal5 and Verbal7 free
Path diagram
End of problem

(Note: You should first establish that the data fit a theoretical model before testing for mean differences in the latent variable. Acceptable model-fit indices
for each group, as well as for both groups combined, should be obtained.)

Y102005.indb 260

3/22/10 3:26:24 PM

Multiple Sample, Multiple Group, and Structured Means Models

261

13.3.2 Model Fit
The current example had individual group and combined group model-fit
indices that were acceptable.
Academic Boys
Group Goodness-of-Fit Statistics
Contribution to Chi-Square = 4.15
Standardized RMR = 0.025
Goodness-of-Fit Index (GFI) = 0.99
Nonacademic Boys
Group Goodness-of-Fit Statistics
Contribution to Chi-Square = 5.97
Standardized RMR = 0.042
Goodness-of-Fit Index (GFI) = 0.99
Global Goodness-of-Fit Statistics
Degrees of Freedom = 6
Minimum Fit Function Chi-Square = 10.11 (P = 0.12)
Root Mean Square Error of Approximation (RMSEA) = 0.046
90 Percent Confidence Interval for RMSEA = (0.0 ; 0.095)
P-Value for Test of Close Fit (RMSEA < 0.05) = 0.27
Comparative Fit Index (CFI) = 1.00
13.3.3 Model Estimation and Testing
The LISREL–SIMPLIS computer output reflects the structured mean equations by replacing the CONST term with the mean value for each group in
the measurement equations.
Measurement Equations: Academic Group
READING5 = 262.37 + 1.00*Verbal5, Errorvar .= 50.15, R² = 0.81
(0.84) (6.02)
312.58 8.34
WRITING5 = 258.67 + 0.84*Verbal5, Errorvar. = 36.48, R² = 0.81
(0.70) (0.024) (4.28)
366.96 34.35 8.52
READING7 = 275.71 + 1.00*Verbal7, Errorvar. = 51.72, R² = 0.82
(0.87) (6.62)
317.77 7.82

Y102005.indb 261

3/22/10 3:26:24 PM

262

A Beginner’s Guide to Structural Equation Modeling

WRITING7 = 268.98 + 0.89*Verbal7, Errorvar. = 57.78, R² = 0.76
(0.80) (0.028) (6.05)
338.00 31.95 9.55

Measurement Equations: Nonacademic Group
READING5 = 262.37 + 1.00*Verbal5, Errorvar.= 23.25, R² = 0.87
(0.84) (6.23)
312.58 3.73
WRITING5 = 258.67 + 0.84*Verbal5, Errorvar. = 42.80, R² = 0.72
(0.70) (0.024) (5.64)
366.96 34.35
7.59
READING7 = 275.71 + 1.00*Verbal7, Errorvar. = 65.67, R² = 0.70
(0.87) (9.87)
317.77 6.65
WRITING7 = 268.98 + 0.89*Verbal7, Errorvar. = 67.36, R² = 0.65
(0.80) (0.028) (8.74)
338.00 31.95
7.71

The structured means model is testing the mean latent variable difference, which is indicated by the Mean Vector of Independent Variables. Results
are interpreted based on the knowledge that the mean latent value on
Verbal5 and Verbal7 are set to zero (0) in the first group (academic boys), so
the values reported here are going to indicate that the second group was
either greater than (positive) or less than (negative) the first group on the
latent variables.
The latent variable mean difference value of −13.80 is indicated for the
first latent variable, which indicates the mean difference was less than
the first group, that is, nonacademic boys scored below academic boys on
verbal ability in the 5th grade.
The latent variable mean difference value of –17.31 is indicated for the
second latent variable, which indicates the mean difference was less than
the first group; that is, nonacademic boys scored below academic boys on
verbal ability in the 7th grade.
Overall, nonacademic boys are scoring below academic boys in the 5th
and 7th grades. The latent variable mean differences are divided by their
standard error to yield a one-sample T value, that is, T = −13.80/1.18 =
−11.71 (within rounding error).
Mean Vector of Independent Variables

Y102005.indb 262

Verbal5

Verbal7

−13.80
(1.18)
−11.71

−17.31
(1.24)
−13.99

3/22/10 3:26:24 PM

Multiple Sample, Multiple Group, and Structured Means Models

263

13.4 Summary
In this chapter we have described multiple samples, multiple group, and
structured means modeling to demonstrate the versatility of structural
equation modeling. The first application involved comparing structure
coefficients across samples of data. We referred to this as a Multiple Sample
Model. The second application involved testing the difference between
parameter estimates given multiple groups, for example, different grade
levels, different countries, or different schools. We referred to this as a
Multiple Group Model. Our third application demonstrated how to test
for mean differences between groups on latent variables. We referred to
this as a Structured Means Model. This extends the basic analysis of variance approach where mean differences on observed variables are tested
but, more importantly, mean differences in latent variables can be tested
(Cole, Maxwell, Arvey, & Salas, 1993).
The chapter presented only one example for each of the applications
because a more in depth coverage is beyond the scope of this book.
However, the LISREL software HELP library provides other examples and
can be searched by using keywords to find other software examples and
explanations. The LISREL User Guide is another excellent reference for
other examples of these applications. We now turn our attention to the
next chapter where other SEM applications are presented and discussed.
Exercises
1. Multiple Sample Model
Nursing programs are interested in knowing if their outcomes are
similar from one semester to the next. Two semesters of data were
obtained on how student effort and learning environment predicted
clinical competence in nursing. The regression model is:

eﬀort

comp

learn

Figure 13.4
Nursing Multiple Sample Model.

Y102005.indb 263

3/22/10 3:26:25 PM

264

A Beginner’s Guide to Structural Equation Modeling

Create a LISREL–SIMPLIS program to test whether the regression
coefficients in the model are the same or statistically significantly different for the two semester samples of data. Semester 1 had 250 nurses
and Semester 2 had 205 nurses. (Note: The means and standard deviations were not available, so assume the data is in standardized form
and only use the correlation matrix in your analysis.)
Semester 1 (N = 250)
Clinical
Effort
Learn

Clinical

Effort

Learn

1.0
.28
.23

1.0
.25

1.0

Semester 2 (N = 205)
Clinical
Effort
Learn

Clinical

Effort

Learn

1.0
.21
.16

1.0
.15

1.0

2. Multiple Group Model
Create a LISREL–SIMPLIS program that produces output to determine if path coefficients are statistically significantly different. You
will need the LISREL–SIMPLIS software and separate data set information provided below to perform this task. Also, provide the path
diagrams with interpretation of results using the Excel program.
The path model tests that job satisfaction (satis) is indicated by boss
attitude (boss) and the number of hours worked (hrs). The boss attitude (boss) is in turn indicated by the employee satisfaction (satis).
The boss attitude (boss) is also indicated by the type of work performed (type), level of assistance provided (assist), and evaluation of
the work (eval). The Equation command would therefore be specified
as follows:
Equation:
satis = boss hrs
boss = type assist eval satis
( Note : Since a reciprocal relation exists between boss and satis,

the errors would need to be correlated to obtain the correct path
coefficients.)

Y102005.indb 264

3/22/10 3:26:25 PM

Multiple Sample, Multiple Group, and Structured Means Models

265

The data set information to be used to test hypotheses of equal or
unequal parameter estimates in a path model between Germany and
the United States are listed below.

Germany
Path Model Analysis for Germany
Observed Variables satis boss hrs type assist eval
Sample Size = 400
Means 1.12 2.42 10.34 4.00 54.13 12.65
Standard Deviation 1.25 2.50 3.94 2.91 9.32 2.01
Correlation Matrix
1.00
.55 1.00
.49 .42 1.00
.10 .35 .08 1.00
.04 .46 .18 .14 1.00
.01 .43 .05 .19 .17 1.00

United States
Path Model Analysis for United States
Observed Variables satis boss hrs type assist eval
Sample Size = 400
Means: 1.10 2.44 8.65 5.00 61.91 12.59
Standard Deviations: 1.16 2.49 4.04 4.41 4.32 1.97
Correlation Matrix
1.00
.69 1.00
.48 .35 1.00
.02 .24 .11 1.00
.11 .19 .16 .31 1.00
.10 .28 .13 .26 .18 1.00

3. Structured Means Model
A researcher is interested in testing whether a low-motivation
group and a high-motivation group in two different cities (Los
Angeles and Chicago) have a production rate mean difference
on the production line. Create and run the two stacked LISREL–
SIMPLIS programs for a test of latent variable mean differences.
Explain results.
The structured means model is diagrammed in Figure 13.5.

Y102005.indb 265

3/22/10 3:26:25 PM

266

A Beginner’s Guide to Structural Equation Modeling

Prod1

Prod2

City 1

Prod3

Prod4

Prod5

City 2

Prod6
Figure 13.5
Motivation Structured Means Model.

The Low-Motivation data information is:
Group Low Motivation:
Observed Variables: Prod1 Prod2 Prod3 Prod4 Prod5 Prod6
Correlation Matrix
1.00
.64 1.00
.78 .73 1.00
.68 .63 .69 1.00
.43 .55 .50 .59 1.00
.65 .63 .67 .81 .60 1.00
Means 4.27 5.02 4.48 4.69 4.53 4.66
Sample Size: 300
The High-Motivation data information is:
Group High Motivation:
Correlation Matrix
1.00
.72 1.00
.76 .74 1.00
.51 .46 .57 1.00
.32 .33 .39 .40 1.00
.54 .45 .60 .73 .45 1.00
Means 14.35 14.93 14.59 14.86 14.71 14.74
Sample Size: 300

Y102005.indb 266

3/22/10 3:26:25 PM

Multiple Sample, Multiple Group, and Structured Means Models

267

Suggested Readings
Multiple Samples
Geary, D. C. & Whitworth, R. H. (1988). Dimensional structure of the Wais-R:
A simultaneous multi-sample analysis. Educational and Psychological
Measurement, 48(4), 945–956.
Tschanz, B. T., Morf, C. C., & Turner, C. W. (1998). Gender differences in the structure of Narcissism: A multi-sample analysis of the narcissistic personality
inventory. Sex Roles: A Journal of Research, 38, 863–868.
Poon, W. Y., & Tang, F. C. (2002). Multisample analysis of multivariate ordinal categorical variables. Multivariate Behavioral Research, 37, 479–500.

Multiple Group Models
Conner, B. T., Stein, J. A., Longshore, D. (2005). Are cognitive AIDS risk-reduction
model equally applicable among high- and low-risk seekers? Personality &
Individual Differences, 38, 379–393.
Long, B. (1998). Coping with workplace stress: A multiple-group comparison of
female managers and clerical workers. Journal of Counseling Psychology, 45,
65–78.
Unrau, N. & Schlackman, J. (2006, November/December). Motivation and its relationship with reading achievement in an urban middle school. The Journal of
Educational Research, 100(2), 81–101.

Structured Means Models
Anderson, N., & Lievens, F. (2006). A construct-driven investigation of gender differences in a leadership-role assessment center. Journal of Applied Psychology,
91, 555–566.
Hancock, G. (2001). Effect size, power, and sampling size determination for structured means modeling and mimic approaches to between groups hypothesis testing of means on a testing of means on a single latent construct.
Psychometrika, 66, 3, 373–388.
Hayashi, N., Igarashi, Y., Yamashina, M., & Suda, K. (2002, January/February). Is
there a gender difference in a factorial structure of the positive and negative
syndrome scale? Psychopathology, 35(1), 28–35.
Wei, M. F., Russell, D. W., Mallinckrodt, B., & Zakalik, R. A. (2004). Cultural
equivalence of adult attachment across four ethnic groups: factor structure,
structured means, and associations with negative mood. Journal of Counseling
Psychology, 51, 408–417.

Y102005.indb 267

3/22/10 3:26:26 PM

268

A Beginner’s Guide to Structural Equation Modeling

Chapter Footnote
SPSS
Select Cases: Random Sample
This dialog box allows you to select a random sample based on an
approximate percentage or an exact number of cases. Sampling is performed without replacement; so, the same case cannot be selected more
than once.
Approximately: Generates a random sample of approximately the
specified percentage of cases. Since this routine makes an independent pseudo-random decision for each case, the percentage of
cases selected can only approximate the specified percentage. The
more cases there are in the data file, the closer the percentage of
cases selected is to the specified percentage.
Exactly: A user-specified number of cases. You must also specify the
number of cases from which to generate the sample. This second
number should be less than or equal to the total number of cases
in the data file. If the number exceeds the total number of cases in
the data file, the sample will contain proportionally fewer cases
than the requested number.
From the menu choose:
Data
Select Cases
Select Random sample of cases.
Click Sample.
Select the sampling method and enter the percentage or number
of cases.

Y102005.indb 268

3/22/10 3:26:26 PM

Multiple Sample, Multiple Group, and Structured Means Models

269

References
Arbuckle, J. L., & Wothke, W. (2003). Amos 5.0 user’s guide. Chicago, IL:
Smallwaters Corporation.
Cole, D. A., Maxwell, S. E., Arvey, R., & Salas, E. (1993). Multivariate group comparisons of variable systems: MANOVA and structural equation modeling.
Psychological Bulletin, 114, 174–184.
Jöreskog, K., & Sörbom, D. (1993). LISREL 8: Structural equation modeling with the
SIMPLIS command language. Chicago, IL: Scientific Software International.
Lomax, R. G. (1985). A structural model of public and private schools. Journal of
Experimental Education, 53, 216–226.
Marcoulides, G., & Schumacker, R. E. (Eds.). (1996). Advanced structural equation
modeling: Issues and techniques. Mahwah, NJ: Lawrence Erlbaum.
Marcoulides, G., & Schumacker, R. E. (Eds.). (2001). New developments and techniques in structural equation modeling: Issues and techniques. Mahwah, NJ:
Lawrence Erlbaum.

Y102005.indb 269

3/22/10 3:26:26 PM

14
Second-Order, Dynamic, and
Multitrait Multimethod Models

Key Concepts
Second-order factors
Dynamic models: measuring factors over time
Establishing reliability and validity when measuring multiple traits
and methods
In the previous chapter we learned about comparing samples or groups
using structural equation modeling applications. In this chapter we
present additional applications that expand our understanding of SEM
models, but now related to measurement models. Please be aware that
our discussion will only scratch the surface of the many exciting new
developments in structural equation modeling related to measurement
models. Some of these new applications have been included in chapters
of books (Marcoulides & Schumacker, 1996; Marcoulides & Schumacker,
2001; and Schumacker & Marcoulides, 1998) and journal articles. In
addition, the newest version of LISREL has included these capabilities
with software examples and further explanations. Our intention is to
provide a basic understanding of these topics to further your interest in
the structural equation modeling approach. We have included computer
program examples to better illustrate each type of SEM model.

14.1 Second-Order Factor Model
14.1.1 Model Specification and Identification
A second-order factor model is indicated when first-order factors are explained
by some higher-order factor structure. Theory plays an important role in justifying a higher-order factor. Visual, verbal, and speed are three psychological factors
271

Y102005.indb 271

3/22/10 3:26:27 PM

272

A Beginner’s Guide to Structural Equation Modeling

visual

ability

verbal

speed

visperc

err_v

cubes

err_c

lozenges

err_l

paragrap

err_p

sentence

err_s

wordmean

err_w

addition

err_p

countdot

err_s

s-c caps

err_w

Figure 14.1
Second-Order Factor Model.

that most likely indicate a second-order factor, namely Ability. A second-order
factor model is therefore hypothesized and diagrammed in Figure 14.1.
14.1.2 Model Estimation and Testing
The data used for testing the second-order factor model is based on an
example in the LISREL 8 Student Examples, SPLEX folder (EX5.spl). The
data are nine psychological variables that identified three common factors
(Visual, Verbal, and Speed). The second-order factor model hypothesizes that
these three common factors indicate a higher-order second factor, Ability.
The LISREL–SIMPLIS program includes the Ability latent variable and
sets the variance of this higher-order second factor to 1.0. (Note: S-C CAPS
loads on both latent variables Visual and Speed, and a single quote is used
when variable names have a space between them.) The LISREL–SIMPLIS
program is therefore written as:
Second-Order Factor Analysis (EX5.SPL)
Observed Variables
‘VIS PERC’ CUBES LOZENGES ‘PAR COMP’ ‘SEN COMP’ WORDMEAN

Y102005.indb 272

3/22/10 3:26:27 PM

Second-Order, Dynamic, and Multitrait Multimethod Models

273

ADDITION COUNTDOT ‘S-C CAPS’
Correlation Matrix
1.000
.318 1.000
.436 .419 1.000
.335 .234 .323 1.000
.304 .157 .283 .722 1.000
.326 .195 .350 .714 .685 1.000
.116 .057 .056 .203 .246 .170 1.000
.314 .145 .229 .095 .181 .113 .585 1.000
.489 .239 .361 .309 .345 .280 .408 .512 1.000
Sample Size 145
Latent Variables: Visual Verbal Speed Ability
Relationships:
‘VIS PERC’ - LOZENGES ‘S-C CAPS’ = Visual
‘PAR COMP’ - WORDMEAN = Verbal
ADDITION - ‘S-C CAPS’ = Speed
Visual = Ability
Verbal = Ability
Speed = Ability
Set variance of Ability = 1.0
Number of Decimals = 3
Wide Print
Print Residuals
Path diagram
End of problem

The selected LISREL–SIMPLIS model-fit indices listed below indicated that
the hypothesized second-order factor model has an acceptable fit (c2 =
28.744, p = .189, df = 23; RMSEA = .04; GFI = .958).
Goodness-of-Fit Statistics
Degrees of Freedom = 23
Minimum Fit Function Chi-Square = 29.008 (P = 0.180)
Normal Theory Weighted Least Squares Chi-Square = 28.744
(P = 0.189)
Estimated Noncentrality Parameter (NCP) = 5.744
90 Percent Confidence Interval for NCP = (0.0; 23.597)
Root Mean Square Error of Approximation (RMSEA) = 0.0416
90 Percent Confidence Interval for RMSEA = (0.0; 0.0844)
P-Value for Test of Close Fit (RMSEA < 0.05) = 0.580
Root Mean Square Residual (RMR) = 0.0451
Standardized RMR = 0.0451
Expected Cross-Validation Index (ECVI) = 0.505
90 Percent Confidence Interval for ECVI = (0.465; 0.629)
ECVI for Saturated Model = 0.625
ECVI for Independence Model = 4.695
Normed Fit Index (NFI) = 0.956
Goodness-of-Fit Index (GFI) = 0.958

Y102005.indb 273

3/22/10 3:26:27 PM

274

A Beginner’s Guide to Structural Equation Modeling

The structural equations in the computer output indicate the strength of
relationship between the first-order factors and the second-order factor,
Ability. Visual (.987) is indicated as a stronger measure of Ability, followed
by Verbal (.565) and Speed (.395), with all three being statistically significant
(t > 1.96). Therefore, student Ability is predominantly a function of visual
perception of geometric configurations with complementary verbal skills
and speed in completing numerical tasks, which enhance a students’
overall ability.
Structural Equations
Visual = 0.987*Ability, Errorvar.= 0.0257, R² = 0.974
(0.229) (0.401)
4.309 0.0640
Verbal = 0.565*Ability, Errorvar.= 0.681 , R² = 0.319
(0.141) (0.170)
4.015 3.997
Speed = 0.395*Ability, Errorvar.= 0.844 , R² = 0.156
(0.132) (0.227)
2.999 3.717

14.2 Dynamic Factor Model
A class of SEM applications that involve stationary and nonstationary latent
variables across time with lagged (correlated) measurement error has been
called dynamic factor analysis (Hershberger, Molenaar, & Corneal, 1996).
A characteristic of the SEM dynamic factor model is that the same measurement instruments are administered to the same subject on two or more
occasions. The purpose of the analysis is to assess change in the latent variable between the ordered occasions due to some event or treatment. When
the same measurement instruments are used over two or more occasions,
there is a tendency for the measurement errors to correlate (autocorrelation);
for example, a specific sequence of correlated error, where error at Time 1
correlates with error at Time 2, and error at Time 2 correlates with error at
Time 3, is called an ARIMA model in econometrics.
Educational research has indicated that anxiety increases the level of
student achievement and performance. Psychological research in contrast
indicates that anxiety has a negative effect upon individuals, thus should
interfere or have a decreasing impact on the level of achievement and performance. Is it possible that both areas of research are correct?
A dynamic factor model was hypothesized to indicate student achievement and performance measures at three equal time points two weeks

Y102005.indb 274

3/22/10 3:26:27 PM

275

Second-Order, Dynamic, and Multitrait Multimethod Models

apart (time 1, time 2, and time 3). The student data indicates achievement
(A1) and performance (P1) at time 1, achievement (A2) and performance
(P2) at time 2, and achievement (A3) and performance (P3) at time 3. The
errors at time 1 were hypothesized to correlate with errors at time 2 and
errors at time 2 were hypothesized to correlate with errors at time 3, indicating an ARIMA model. Time 1 predicts time 2 and time 2 predicts time
3. The dynamic factor model is diagrammed in Figure 14.2a:

error

Time 2

Time 1

error

Time 3

error

Figure 14.2a
Dynamic Factor Model (Wheaton et al., 1977).

The data set contains 600 students who were measured on the same
achievement and performance measures at three different points in time.
The two variables, achievement and performance, defined the factor time.
Thus, the latent variable, time, was represented as time1, time2, and time3,
with two indicator variables at each time point. How well students did
at time 2 was predicted by the time 1 latent variable. Likewise, how well
students did at time 3 was predicted by time 2. Students were given a high
level of anxiety by having to meet deadlines, take frequent quizzes, and
turn in extra assignments. A LISREL–SIMPLIS program was created to
test this dynamic factor model.
Dynamic Factor Model
Observed Variables: A1 P1 A2 P2 A3 P3
Covariance Matrix
11.834
6.947 9.364
6.819 5.091 12.532

Y102005.indb 275

3/22/10 3:26:28 PM

276

A Beginner’s Guide to Structural Equation Modeling

4.783 5.028 7.495 9.986
−3.839 −3.889 −3.841 −3.625 9.610
−2.190 −1.883 −2.175 −1.878 3.552 4.503
Sample Size: 600
Latent Variables: Time1 Time2 Time3
Relationships:
A1 P1 = Time1
A2 P2 = Time2
A3 P3 = Time3
Time2 = Time1
Time3 = Time2
Let the Errors of A1 and A2 correlate
Let the Errors of P1 and P2 correlate
Let the Errors of A2 and A3 correlate
Let the Errors of P2 and P3 correlate
Let the Errors of Time2 and Time3 correlate
Path Diagram
End of Problem

The dynamic factor model results indicated an acceptable model fit (c2 =
2.76, df = 2, and p = .25). The structural equations indicate the prediction
across the three time intervals for the latent variable, time. Time 1 was
a statistically significant predictor of time 2; coefficient was statistically
significant (T = 12.36) and R 2 = .47. Time 2 was a statistically significant
predictor of time 3; however, the result indicated a negative coefficient
(−.82).
Structural Equations
Time2 = 0.68*Time1, Errorvar. = 0.53 , R² = 0.47
(0.055) (0.071)
12.36 7.50
Time3 = − 0.82*Time2, Errorvar. = 0.80 , R² = 0.20
(0.085)
(0.12)
−9.66 6.52

The dynamic factor model would therefore be interpreted as follows: anxiety increased the level of student achievement and performance from time
1 to time 2, but then decreased the level of student achievement and performance from time 2 to time 3. Anxiety increased levels of achievement and
performance, but only for a certain amount of time, then it had a negative
effect. So, it appears educational researchers and psychologists are both
correct to some extent. The dynamic factor model clarifies how anxiety
affects the level of student achievement and performance, given a time
continuum.
The dynamic factor model output with standardized coefficients is
listed in Figure 14.2b with standardized coefficients:

Y102005.indb 276

3/22/10 3:26:28 PM

277

Second-Order, Dynamic, and Multitrait Multimethod Models

1.54

0.41

0.16

0.004

error

Time 1

0.68

–0.82

Time 2

error

Time 3

error

0.29
Figure 14.2b
Dynamic Factor Model Output.

14.3 Multitrait Multimethod Model (MTMM)
The Multitrait Multimethod model (MTMM) is used to indicate multiple
traits assessed by multiple measures—for example, student achievement
and student motivation (traits) assessed by teacher ratings and student self
ratings (methods). MTMM models, however, are problematic to analyze
as noted by Lomax and Algina (1979) who compared two procedures for
analyzing MTMM matrices. The MTMM matrix does provide a convenient
way to report reliability and construct validity coefficients (Campbell &
Fiske, 1959). Construct validity involves providing psychometric evidence
of convergent validity, discriminant validity, trait, and method effects, even
across populations (Cole & Maxwell, 1985). The Multitrait Multimethod
matrix conveniently displays the convergent validity coefficients, discriminant validity coefficients, and the reliability coefficients along the diagonal. A two-trait/two-method matrix is displayed in Table 14.1.
Reliability coefficients (1) indicate the internal consistency of scores on
the instrument, and therefore should be in the range .85 to .95 or higher.
Convergent validity coefficients (2) are correlations between measures
of the same trait (construct) using different methods (instruments), and
therefore should also be in the range .85 to .95 or higher. Discriminant
validity coefficients (3) are correlations between measures of different

Y102005.indb 277

3/22/10 3:26:29 PM

278

A Beginner’s Guide to Structural Equation Modeling

Table 14.1
Two-Trait/Two-Method Multitrait Multimethod
Matrix
Method 1 Method 2
Trait
Method 1. Self Ratings
A. Achievement
B. Motivation
Method 2. Teacher Ratings
A. Achievement
B. Motivation

(1)
(3)

(1)

(2)
—

—
(2)

(1)
(3)

(1)

Note: (1) = reliability coefficients; (2) = convergent validity coefficients; (3) = discriminant validity coefficients; and (—) = correlations between ratings that
share neither trait nor method.

traits (constructs) using the same method (instrument), and should be
much lower than the convergent validity coefficients and/or the instrument reliability coefficients. The basic MTMM model for two traits/two
methods is diagrammed in Figure 14.3a:
Rating 1
Achieve
F1
Rating 2

Rating 3
Motivate
F2
Rating 4

Self
Rating
F3

Teacher
Rating
F4

Figure 14.3a
Basic MTMM Model (two traits/two methods).

The correlation of ratings from different methods of the same trait
should be statistically significant—that is, having convergent validity
(2). The convergent validity coefficients should also be greater than the
correlations of ratings from different traits using the same method—that
is, discriminant validity (3), and the correlations between ratings that
share neither trait nor method (−).

Y102005.indb 278

3/22/10 3:26:29 PM

Second-Order, Dynamic, and Multitrait Multimethod Models

279

14.3.1 Model Specification and Identification
The Multitrait Multimethod (MTMM) model example indicates three
methods (self-ratings, peer ratings, and observer ratings) used to assess
four traits of leadership (prominence, achievement, affiliation, and leader).
The sample size is N = 240 subjects.
MTMM models are problematic to analyze and typically will require
specifying start values (initial parameter values) and setting AD = OFF
(admissibility check) to obtain convergence—that is, obtain parameter
estimates. (Note: Start values are typically chosen between .1 and .9 so that
the estimation process does not have to start with a zero value for parameters in the model; the 2SLS estimates also provide reasonable start values). The Multitrait Multimethod models are difficult to analyze because
they lack model identification (initially have negative degrees of freedom) and can have convergence problems (nonpositive definite matrix).
Consequently, latent variable variances should be set to 1.0, and factor
correlations between traits and methods set to zero, otherwise, the PHI
matrix will be nonpositive definite. Additionally, certain error variances
need to be set equal to prevent negative error variance (Heywood case).
In MTMM models, the different methods are uncorrelated with the different traits, so a model diagram helps to visually display the specified model
(Figure 14.3b).
X1
prom

self

X3
X4
ach

X5
X6

peer

X7
affl

X8
X9
X10

lead

obs

X11
X12

Figure 14.3b
Multitrait Multimethod Model. (From Bollen, K. A. [1989]. Structural equations with latent
variables. New York: John Wiley & Sons.)

Y102005.indb 279

3/22/10 3:26:30 PM

280

A Beginner’s Guide to Structural Equation Modeling

14.3.2 Model Estimation and Testing
The LISREL–SIMPLIS program to analyze the three sets of ratings on the
four traits as a MTMM model is:
Multitrait-Multimethod Bollen (1989)
!Start Values Added (.5) and Admissibility Check Off (AD=OFF)
Observed Variables: X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12
Correlation Matrix
1.0
.50 1.0
.41 .48 1.0
.67 .59 .40 1.0
.45 .33 .26 .55 1.0
.36 .32 .31 .43 .72 1.0
.25 .21 .25 .30 .59 .72 1.0
.46 .36 .28 .51 .85 .80 .69 1.0
.53 .41 .34 .56 .71 .58 .43 .72 1.0
.50 .45 .29 .52 .59 .55 .42 .63 .84 1.0
.36 .30 .28 .37 .53 .51 .43 .57 .62 .57 1.0
.52 .43 .31 .59 .68 .60 .46 .73 .92 .89 .63 1.0
Sample Size: 240
Latent Variables: prom ach affl lead self peer obs
Relationships:
X1 = (.3)*self + (.5)*prom
X2 = (.3)*self + (.5)*ach
X3 = (.3)*self + (.5)*affl
X4 = (.3)*self + (.5)*lead
X5 = (.3)*peer + (.5)*prom
X6 = (.2)*peer + (.5)*ach
X7 = (.2)*peer + (.5)*affl
X8 = (.2)*peer + (.5)*lead
X9 = (.2)*obs + (.5)*prom
X10 = (.3)*obs + (.5)*ach
X11 = (.3)*obs + (.5)*affl
X12 = (.3)*obs + (.5)*lead
Set Variance of prom - obs to 1.0
Set correlation of prom and self to 0
Set correlation of ach and self to 0
Set correlation of affl and self to 0
Set correlation of lead and self to 0
Set correlation of prom and peer to 0
Set correlation of ach and peer to 0
Set correlation of affl and peer to 0
Set correlation of lead and peer to 0
Set correlation of prom and obs to 0
Set correlation of ach and obs to 0
Set correlation of affl and obs to 0

Y102005.indb 280

3/22/10 3:26:30 PM

281

Second-Order, Dynamic, and Multitrait Multimethod Models

Set correlation of lead and obs to 0
Let the error variance of X10 and X12 be equal
OPTIONS: AD = OFF
LISREL OUTPUT
End of Problem

Results from the computer output are summarized in Table 14.2 to demonstrate the interpretation of trait and method effects. The assessment
of Affiliation (Affl) had the highest error variance when using Self ratings
(error = .67) and Observer ratings (error = .39), thus Affiliation was the most
difficult trait to assess using either of these two methods. The self rating
worked best for leadership assessment (factor loading = .61; error variance =
.30). The peer rating method worked best with assessing affiliation (factor
loading = .79; error variance = .14). The observer rating method worked best
with assessing achievement (factor loading = .68; error variance = .07).
Most attempts at running MTMM models will result in unidentified models or lack convergence (unable to estimate parameters). Other
types of MTMM models—for example, correlated uniqueness model or
a composite direct product model—generally work better. A correlated
uniqueness model will therefore be presented next.

(Note :

14.3.3 Correlated Uniqueness Model
We present an example of a correlated uniqueness model, since it seems
to have less convergence problems with meaningful results, and is
Table 14.2
MTMM Estimates of Four Traits Using Three Methods (N = 240)
Traits
Prom
Prom
Ach
Affl
Lead

.52

Prom
Ach
Affl
Lead

.84

Prom
Ach
Affl
Lead

.80

Y102005.indb 281

Ach

Methods
Affl

Lead

Self

.58

.58
.61
.47
.61

.42
.35

.69
.48
.84

.69
.75
.78

Peer

Obs

Error
.41
.46
.67
.30

.32
.53
.79
.43

.19
.23
.14
.09
.53
.68
.23
.59

.09
.07
.39
.07

3/22/10 3:26:30 PM

282

A Beginner’s Guide to Structural Equation Modeling

recommended by Marsh and Grayson (1995) and Wothke (1996) as an
alternative to traditional MTMM models. In correlated uniqueness models, each variable is affected by one trait factor and one error term, and
there are no method factors. The method effects are accounted for by the
correlated error terms of each variable. The correlated error terms only
occur between variables measured by the same method.
Different types of correlated uniqueness models can be analyzed
(Huelsman, Furr, & Nemanick, 2003). For example, one general factor with
correlated uniqueness, two correlated factors with correlated uniqueness,
two correlated factors with uncorrelated uniqueness, or two uncorrelated
factors with correlated uniqueness. Marsh and Grayson (1995) indicated
that a significant decrease in fit between a model with correlated traits,
but no correlated error terms and a model with correlated traits and correlated error terms, indicated the presence of method effects. Following
this approach, you can test method effects by analyzing a correlated trait
correlated uniqueness model (CTCU) and a correlated trait (CT) only
model.
Figure 14.3c displays the correlated trait–correlated uniqueness (CTCU)
model with three traits and three methods. The CTCU model represents
the method effects through the correlated error terms of the observed
variables. Figure 14.3d displays the correlated trait (CT) only model with
no correlated error terms. In the CT model, the variables measured by the
same method are grouped under each trait factor.

Trait 1

Trait 2

Trait 3

Var 1

Var 2

Var 3

Var 4

Var 5

Var 6

Var 7

Var 8

Var 9

Figure 14.3c
Correlated Trait–Correlated Uniqueness Model.

Y102005.indb 282

3/22/10 3:26:31 PM

283

Second-Order, Dynamic, and Multitrait Multimethod Models

Trait 1

Trait 2

Trait 3

Var 1

Var 2

Var 3

Var 4

Var 5

Var 6

Var 7

Var 8

Var 9

Figure 14.3d
Correlated Trait Model.

The data from Bollen (1989) was used again, but this time only three
traits (prom, ach, and affl) with three methods (self, peer, and obs) were analyzed. The LISREL–SIMPLIS program for the CTCU model with correlated traits and correlated error terms is:
Correlated Traits—Correlated Uniqueness Model - Bollen (1989)
Observed Variables: Var1 Var2 Var3 Var4 Var5 Var6 Var7
Var8 Var9
Correlation Matrix
1.0
.50 1.0
.41 .48 1.0
.45 .33 .26 1.0
.36 .32 .31 .72 1.0
.25 .21 .25 .59 .72 1.0
.53 .41 .34 .71 .58 .43 1.0
.50 .45 .29 .59 .55 .42 .84 1.0
.36 .30 .28 .53 .51 .43 .62 .57 1.0
Sample Size: 240
Latent Variables: prom ach affl
Relationships:
Var1 = prom
Var2 = ach
Var3 = affl
Var4 = prom
Var5 = ach
Var6 = affl

Y102005.indb 283

3/22/10 3:26:31 PM

284

A Beginner’s Guide to Structural Equation Modeling

Var7 = prom
Var8 = ach
Var9 = affl
Set Variance of prom-affl to
Let Error Covariance of Var1
Let Error Covariance of Var4
Let Error Covariance of Var7
Path Diagram
End of Problem

1.0
– Var3 Correlate
– Var6 Correlate
– Var9 Correlate

The results from the computer output are presented in Table 14.3. Findings
indicated that all three traits were statistically significantly correlated.
More importantly, the observation method was the best for assessing any of
the three traits, as indicated by the higher trait factor loadings and lower
correlated uniqueness error terms. The data also had an acceptable fit to
the CTCU model (c2 = 17.38, p = .30, df = 15; RMSEA = .026).
The LISREL program was run again to estimate a correlated trait (CT)only model with no correlated error terms. To accomplish this, you simply
delete the following command lines:
Let Error Covariance of Var1 – Var3 Correlate
Let Error Covariance of Var4 – Var6 Correlate
Let Error Covariance of Var7 – Var9 Correlate

Table 14.3
Correlated Uniqueness Model with Correlated Traits and Errors
Method
Self

Peer

Observe

Trait
Prom
Ach
Affl
Prom
Ach
Affl
Prom
Ach
Affl

Trait correlations
Prom
1.0
Ach
.93
Affl
.88

Factor
Loading

Uniqueness

Correlated Uniqueness
of Error Terms

.58
.48
.40
.78
.68
.55
.92
.84
.76

.67
.77
.85
.40
.54
.70
.16
.30
.42

.33
.23
.16
.61
.46
.30
.84
.70
.58

1.0
.24
.20
1.0
.23
.23
1.0
.12
.007

1.0
.93

1.0

1.0
.30

1.0

1.0
.37

1.0

1.0
−.03

1.0

Note: c = 17.38, p = .30, df = 15; RMSEA = .026; n = 240.
2

Y102005.indb 284

3/22/10 3:26:31 PM

285

Second-Order, Dynamic, and Multitrait Multimethod Models

Table 14.4
Correlated Uniqueness Model with Correlated Traits Only
Method

Trait

Self

Prom
Ach
Affl
Prom
Ach
Affl
Prom
Ach
Affl

Peer

Observe

Trait Correlationsa
Prom
1.0
Ach
1.05
Affl
.95

Factor Loading

Uniqueness

.58
.45
.41
.79
.72
.62
.90
.80
.68

.66
.79
.83
.37
.48
.61
.20
.35
.53

.34
.21
.17
.63
.52
.39
.80
.65
.47

1.0
1.06

1.0

Note: c2 = 270.63, p = .0000, df = 24; RMSEA = .21; n = 240.
Trait correlation matrix is a nonpositive definite matrix because correlations
are greater than 1.0.

The results from the computer output are presented in Table 14.4. The
trait factor loadings, uniqueness, and R2 values are not substantially different from the previous CTCU model; however, the data is not an acceptable fit to the CT model (c2 = 270.63, p = .00001, df = 24; RMSEA = .21).
Comparing the previous CTCU model (c2 = 17.38, p = .30) to this CT model
(c2 = 270.63, p = .00001) indicates a method effect. The method that was
suggested as more effective was the observation method. Some trait correlations in the CT model were greater than 1.0 indicating a nonpositive
definite matrix (1.05 and 1.06—boldfaced). The CT model modification
indices also suggested adding the specific unique error covariance terms
which, if added, would result in the CTCU model.
The Modification Indices Suggest to Add an Error Covariance
Between and Decrease in Chi-Square New Estimate
Var2 Var1 21.4 0.22
Var3 Var1 15.4 0.19
Var3 Var2 27.8 0.28
Var5 Var4 30.4 0.19
Var6 Var4 23.0 0.17
Var6 Var5 76.3 0.35
Var7 Var5 41.0 −0.21
Var7 Var6 27.5 −0.16

Y102005.indb 285

3/22/10 3:26:32 PM

286

A Beginner’s Guide to Structural Equation Modeling

Var8 Var4 33.7 −0.18
Var8 Var6 10.3 −0.12
Var8 Var7 70.3 0.26

The MTMM model is problematic to analyze, but can be done given
the addition of start values, setting AD = OFF, setting latent variances
to 1.0, setting factor correlations to zero, and setting error variances
equal. The alternative correlated uniqueness model approach in SEM
is easier to obtain convergence (compute parameter estimates), but is
not without controversy over how to interpret the results because more
than one possible explanation may exist for the observed correlated
error terms.
Although Bollen (1989, p. 190–206) and Byrne (1998, p. 228–229) have
demonstrated how to conduct a multitrait multmethod model with a
taxonomy of nested models suggested by Widaman (1985), Marsh and
Grayson (1995) and Wothke (1996) have demonstrated that most attempts
at running MTMM models result in unidentified models or lack convergence, and offer suggestions for other types of MTMM models that
included the correlated uniqueness model or a composite direct product
model. We strongly suggest that you read Marsh and Grayson (1995) or
Wothke (1996) for a discussion of these alternative MTMM models and
problems with analyzing data using a MTMM model.
Saris and Aalberts (2003) questioned the interpretation of the correlated uniqueness model approach in SEM. They agreed that one
possible explanation for the observed correlated terms is the similarity of methods for the different traits; however, they provided other
explanations for the correlated error terms. Their alternative models
explained the correlated error terms based on method effects, relative
answers to questions, acquiescence bias, and/or variation in response
patterns when examining characteristics of survey research questions
on a questionnaire. We are, therefore, reminded that error terms do
not necessarily reflect a single unknown measure, rather contain sampling error, systematic error, and other potentially unknown measures
(observed variables).

14.4 Summary
In this chapter, we have considered second-order factor models, dynamic
factor models, and multitrait multimethod models, including an alternative correlated uniqueness model. We have learned that the traditional

Y102005.indb 286

3/22/10 3:26:32 PM

Second-Order, Dynamic, and Multitrait Multimethod Models

287

multitrait multimethod model has identification and convergence problems such that Marsh and Grayson (1995) and Wothke (1996) have recommended alternative approaches, namely correlated uniqueness and direct
product models.
We hope that our discussion of these SEM applications has provided
you with a basic overview and introduction to these methods. We encourage you to read the references provided at the end of the chapter and run
some of the program setups provided in the chapter. We further hope
that the basic introduction in this chapter will permit you to read the
research literature and better understand the resulting models presented,
which should support various theoretical perspectives. Attempting a few
basic models will help you better understand the approach; afterwards,
you may wish to attempt one of these SEM applications in your own
research.
Exercises
1. Second-Order Factor Analysis
The psychological research literature tends to suggest that drug use
and depression are leading indicators of suicide among teenagers.
(Note: Set variance of Suicide = 1 for model identification purposes).
Given the following data set information, create and run a LISREL–
SIMPLIS program to conduct a second-order factor analysis.

Observed Variables: drug1 drug2 drug3 drug4 depress1
depress2 depress3 depress4
Sample Size 200
Correlation Matrix
1.000
0.628 1.000
0.623 0.646 1.000
0.542 0.656 0.626 1.000
0.496 0.557 0.579 0.640 1.000
0.374 0.392 0.425 0.451 0.590 1.000
0.406 0.439 0.446 0.444 0.668 .488 1.000
0.489 0.510 0.522 0.467 0.643 .591 .612 1.000
Means 1.879 1.696 1.797 2.198 2.043 1.029 1.947 2.024
Standard Deviations 1.379 1.314 1.288 1.388 1.405 1.269
1.435 1.423
Latent Variables: drugs depress Suicide

The second-order factor model is diagrammed in Figure 14.4:

Y102005.indb 287

3/22/10 3:26:32 PM

288

A Beginner’s Guide to Structural Equation Modeling

drug1
d1
drug2
drugs

drug3
drug4

Suicide
depress1

depress2
depress
depress3
d2
depress4
Figure 14.4
Siucide Second-Order Factor Model.

2. Dynamic Factor Model
A sports physician was interested in studying heart rate and muscle
fatigue of female soccer players. She collected data after three soccer
games over a 3-week period. A dynamic factor model was used to
determine if heart rate and muscle fatigue were stable across time for
the 150 female soccer players.
Create a LISREL–SIMPLIS program to analyze and interpret the
dynamic factor model. Include a diagram of the dynamic factor
model. The data set information including observed variables, covariance matrix, sample size, and latent variables are provided below:
Observed Variables: HR1 MF1 HR2 MF2 HR3 MF3
Covariance Matrix
10.75
7.00 9.34
7.00 5.00 11.50
5.03 5.00 7.49 9.96
3.89 4.00 3.84 3.65 9.51
2.90 2.00 2.15 2.88 3.55 5.50
Sample Size: 150
Latent Variables: Time1 Time2 Time3

Y102005.indb 288

3/22/10 3:26:32 PM

Second-Order, Dynamic, and Multitrait Multimethod Models

289

3. Multitrait Multimethod (MTMM) Models
Students provided ratings of their classroom behavior, motivation
to achieve, and attitude toward learning. Teachers, likewise, provided ratings of student classroom behavior, perception of students’
motivation to achieve, and attitude toward learning. Finally, other
students or peers provided ratings on these three traits. The three
ratings (student, teacher, and peer) on three traits (behavior, motivate,
attitude) were analyzed in a SEM Multitrait Multimethod model. The
Multitrait Multimethod Model is diagrammed in Figure 14.5:

X1
Student

Behavior

X3
X4
Teacher

Motivate

X6
X7
Peer

Attitude

X9
Figure 14.5
Classroom MTMM Model.

a. Create and run a LISREL–SIMPLIS program to analyze the
three sets of ratings on the three traits as a MTMM model. The
observed variables, correlation matrix, sample size, and latent
variables are:
Observed Variables: X1 X2 X3 X4 X5 X6 X7 X8 X9
Correlation Matrix
1.0
.40 1.0
.31 .38 1.0
.35 .23 .16 1.0
.26 .22 .21 .62 1.0
.15 .11 .15 .49 .62 1.0

Y102005.indb 289

3/22/10 3:26:33 PM

290

A Beginner’s Guide to Structural Equation Modeling

.43 .31 .24 .61 .48 .33 1.0
.40 .35 .19 .49 .45 .32 .74 1.0
.26 .20 .18 .43 .41 .33 .52 .47 1.0
Sample Size: 300
Latent Variables: behavior motivate attitude student
teacher peer

b. Create and run a LISREL–SIMPLIS program to compute a
CTCU and CU model using the data information from above.
Compare the CTCU and CU model results to determine if a
method effect exists. Also, compare the CTCU model with the
MTMM model above, which provides clearer results?

Suggested Readings
Second-Order Factor Models
Chan, D. W. (2006, Fall). Perceived multiple intelligences among male and female
Chinese gifted students in Hong Kong: The structure of the student multiple
intelligences profile. The Gifted Child Quarterly, 50(4), 325–338.
Cheung, D. (2000). Evidence of a single second-order factor in student ratings
of teaching effectiveness. Structural Equation Modeling: A Multidisciplinary
Journal, 7, 442–460.
Rand, D., Conger, R. D., Patterson, G. R., & Ge, X. (1995). It takes two to replicate:
A mediational model for the impact of parents’ stress on adolescent adjustment. Child Development, 66(1), 80–97.

Dynamic Factor Models
Chow, S. M., Nesselroade, J. R., Shifren, K., & McArdle, J. J. (2004). Dynamic structure of emotions among individuals with Parkinson’s disease. Structural
Equation Modeling, 11(4), 560–582.
Kroonenberg, P. M., van Dam, M., van Uzendoorn, M. H., & Mooijaart, A. (1997,
May). Dynamics of behaviour in the strange situation: A structural equation
approach. British Journal of Psychology, 88, 311–332.
Zuur, A. F., Fryer, R. J., Jolliffe, I. T., Dekker, R., Beukema, J. J. (2003). Estimating
common trends in multivariate time series using dynamic factor analysis.
Environmetrics, 14(7), 665–685.

Multitrait Multimethod Models
Bunting, B. P., Adamson, G., & Mulhall, P. K. (2002). A Monte Carlo examination
of an MTMM model with planned incomplete data structures. Structural
Equation Modeling: A Multidisciplinary Journal, 9, 369–389.
Lim, B., & Ployhart, R. E. (2006, January). Assessing the convergent and discriminant validity of Goldberg’s international personality item pool. Organizational
Research Methods, 9(1), 29–54.

Y102005.indb 290

3/22/10 3:26:33 PM

Second-Order, Dynamic, and Multitrait Multimethod Models

291

Tildesley, E. A., Hops, H., Ary, D., & Andrews, J. A. (1995). Multitrait-multimethod
model of adolescent deviance, drug use, academic, and sexual behaviors.
Journal of Psychopathology and Behavioral Assessment, 17(2), 185–215.

Correlated Uniqueness Model
Lievens, F., & Van Keer, E. (2001, September). The construct validity of a Belgian
assessment centre: A comparison of different models. Journal of Occupational
and Organizational Psychology, 74, 373–378.
Marsh, H. W., Roche, R. A., Pajares, F., & Miller, M. (1997). Item-specific efficacy
judgments in mathematical problem solving: The downside of standing too
close to trees in a forest. Contemporary Educational Psychology, 22, 363–377.
Quilty, L. C., Oakman, J. M., & Riski, E. (2006). Correlates of the Rosenberg selfesteem scale method effects. Structural Equation Modeling: A Multidisciplinary
Journal. 13, 99–117.

References
Bollen, K. A. (1989). Structural equations with latent variables. New York: John Wiley
& Sons.
Byrne, B. M. (1998). Structural equation modeling with LISREL, PRELIS, and SIMPLIS:
Basic concepts, applications, and programming. Mahwah, NJ: Lawrence Erlbaum.
Campbell, D. T., & Fiske, D. W. (1959). Convergent and discriminant validation by
the multitrait-multimethod matrix. Psychological Bulletin, 56, 81–105.
Cole, D. A., & Maxwell, S. E. (1985). Multitrait‑multimethod comparisons across
populations: A confirmatory factor analytic approach. Multivariate Behavioral
Research, 20, 389–417.
Hershberger, S. L., Molenaar, P. C. M., & Corneal, S. E. (1996). A hierarchy of univariate and multivariate structural times series models (pp. 159–194). In
Marcoulides, G. & Schumacker, R. E. (Eds.), Advanced structural equation modeling: Issues and techniques. Mahwah, NJ: Lawrence Erlbaum.
Huelsman, T. J., Furr, M. R., Nemanick, Jr., R. C. (2003). Measurement of dispositional affect: Construct validity and convergence with a circumplex model of
affect. Educational and Psychological Measurement, 63(4), 655–673.
Lomax, R. G., & Algina, J. (1979). Comparison of two procedures for analyzing
multitrait multimethod matrices. Journal of Educational Measurement, 16, 177–186.
[errata: 1980, 17, 80]
Marcoulides, G., & Schumacker, R. E. (Eds.). (1996). Advanced structural equation
modeling: Issues and techniques. Mahwah, NJ: Lawrence Erlbaum.
Marcoulides, G., & Schumacker, R. E. (Eds.). (2001). New developments and techniques in structural equation modeling. Mahwah, NJ: Lawrence Erlbaum.
Marsh, H. W., & Grayson, D. (1995). Latent variable models of multitraitmultimethod data. In Hoyle, R. H. (Ed.). Structural equation modeling: Concepts,
issues, and applications. Thousand Oaks, CA: Sage Publications.

Y102005.indb 291

3/22/10 3:26:33 PM

292

A Beginner’s Guide to Structural Equation Modeling

Saris, W. E., & Aalberts, C. (2003). Different explanations for correlated disturbance
terms in MTMM studies. Structural Equation Modeling, 10(2), 193–213.
Schumacker, R. E., & Marcoulides, G. A. (1998). Interaction and nonlinear effects in
structural equation modeling. Mahwah, NJ: Lawrence Erlbaum.
Widaman, K. F. (1985). Hierarchically tested covariance structure models for
multitrait-multimethod data. Applied Psychological Measurement, 9, 1–26.
Wheaton, B., Muthén, B., Alwin, D. F., & Summers, G. F. (1977). Assessing reliability and stability in panel models. In D. R. Heise (Ed.), Sociological methodology
(pp 84–136). San Francisco, CA: Jossey-Bass.
Wothke, W. (1996). Models for multitrait-multimethod matrix analysis. In
Marcoulides, G. & Schumacker, R. E. (Eds.), Advanced structural equation modeling: Issues and techniques (pp. 7–56). Mahwah, NJ: Lawrence Erlbaum.

Y102005.indb 292

3/22/10 3:26:33 PM

15
Multiple Indicator–Multiple Indicator
Cause, Mixture, and Multilevel Models

Key Concepts
Multiple indicator–multiple cause (MIMIC) models
SEM models with continuous and categorical variables (mixture
models)
Testing multilevel intercept and slope differences in nested groups
(multilevel models).
In this chapter we continue with our presentation and discussion of
SEM model applications. Specifically, we present an example where
latent variables are predicted by observed variables (MIMIC model);
an example where continuous and categorical variables are included in
the model (mixture model); and finally an example where nested design
data occur (multilevel model). All three of these SEM applications are
unique and are not possible using traditional statistics (analysis of variance, etc.).

15.1 Multiple Indicator–Multiple Cause (MIMIC) Models
The term MIMIC refers to multiple indicators and multiple causes and
defines a particular type of SEM model. The MIMIC model involves using
latent variables that are predicted by observed variables. An example by
Jöreskog and Sörbom (1996a, example 5.4, p. 185–187) is illustrated where a
latent variable (social participation) is defined by church attendance, memberships, and friends. The social participation latent variable is predicted by
the observed variables, income, occupation, and education. The MIMIC
model is diagrammed in Figure 15.1a.
293

Y102005.indb 293

3/22/10 3:26:33 PM

294

A Beginner’s Guide to Structural Equation Modeling

income

occup

error

social

educ

church

err_c

member

err_m

friends

err_f

Figure 15.1a
MIMIC Model.

The MIMIC model indicates a latent variable, social, which has arrows
pointing out to the three observed indicator variables (church, member,
friends) with separate measurement error terms for each. This is the measurement part of the MIMIC model that defines the latent variable. In the
MIMIC model, the latent variable, social, also has arrows pointed toward it
from the three observed predictor variables, which have implied correlations
among them (curved arrows). This is the structural part of the MIMIC model
that uses observed variables to predict a latent variable. The MIMIC model
diagram also shows the prediction error for the latent variable, social.
15.1.1 Model Specification and Identification
Model identification pertains to whether the estimates in the MIMIC
model can be calculated, which is quickly gauged by the degrees of freedom. Do you recall how the degrees of freedom are determined? There
are a total of 15 free parameters to be estimated in the MIMIC model. The
number of distinct values in the variance-covariance matrix S based on
6 observed variables is: p (p + 1)/2 = 6 (6 + 1)/2 = 21. The degrees of freedom are computed by subtracting the number of free parameters from the
number of distinct parameters in the matrix S, which is 21 − 15 = 6.
15.1.2 Model Estimation and Model Testing
The MIMIC model diagram provides the basis for specifying the LISREL–
SIMPLIS program, specifically the Latent Variable and Relationships command
lines in the LISREL–SIMPLIS program. The LISREL–SIMPLIS program that
specifies the observed variables, sample size, correlation matrix (standardized variables), and the equations that reflect the MIMIC model is:
MIMIC Model
Observed Variables income occup educ church member friends

Y102005.indb 294

3/22/10 3:26:34 PM

Multiple Indicator–Multiple Indicator Cause, Mixture

295

Sample Size 530
Correlation Matrix
1.000
.304 1.000
.305 .344 1.000
.100 .156 .158 1.000
.284 .192 .324 .360 1.000
.176 .136 .226 .210 .265 1.000
Latent Variable social
Relationships
church = social
member = social
friends= social
social = income occup educ
Path Diagram
End of Problem

SEM MIMIC models use goodness-of-fit criteria to determine whether a
reasonably good fit of the data to the MIMIC model exists. Some basic fit
criteria are printed below from the computer output.
Goodness-of-Fit Statistics
Degrees of Freedom = 6
Minimum Fit Function Chi-Square = 12.50 (P = 0.052)
Normal Theory Weighted Least Squares Chi-Square = 12.02
(P = 0.061)
Estimated Noncentrality Parameter (NCP) = 6.02
90 Percent Confidence Interval for NCP = (0.0 ; 20.00)
Root Mean Square Error of Approximation (RMSEA) = 0.044
90 Percent Confidence Interval for RMSEA = (0.0 ; 0.079)
P-Value for Test of Close Fit (RMSEA < 0.05) = 0.56
Expected Cross-Validation Index (ECVI) = 0.079
90 Percent Confidence Interval for ECVI = (0.068 ; 0.11)
Normed Fit Index (NFI) = 0.97
Goodness-of-Fit Index (GFI) = 0.99

The Normal Theory Weighted Least Squares c2 = 12.02, df = 6, and p =
.061 suggests a reasonably good fit of the data to the MIMIC model. The
Goodness-of-Fit (GFI) Index suggests that 99% of the variance-covariance
in matrix S is reproduced by the MIMIC model. The LISREL software standardized solution indicates factor loadings of .47 * church, .74 * member,
and .40 * friends. However, the T-value in the computer output dropped
church as an important indicator variable in defining the latent variable,
social.

Y102005.indb 295

3/22/10 3:26:34 PM

296

A Beginner’s Guide to Structural Equation Modeling

The observed variables, member (T = 6.71) and friends (T = 6.03), were
therefore selected to define the latent variable social. The measurement
equations from the computer output are listed below.
Measurement Equations
church = 0.47*social, Errorvar. = 0.78 , R² = 0.22
(0.058)
13.61
member = 0.74*social, Errorvar. = 0.46 , R² = 0.54
(0.11) (0.075)
6.71 6.10
friends = 0.40*social, Errorvar. = 0.84 , R² = 0.16
(0.067) (0.058)
6.03 14.51

Because a matrix was used rather than raw data, standard error
and T-value are not output for the reference indicator variable, church. The
HELP menu offers this explanation: LISREL for Windows uses a reference
indicator (indicator with a unit factor loading) to set the scale of each of
the endogenous latent (ETA) variables of the model. If you do not specify
reference indicators for the endogenous latent variables of your model,
LISREL for Windows will select a reference indicator for each endogenous
latent variable of your model. Although LISREL for Windows scales the
factor loadings to obtain the appropriate estimates for the factor loadings
of the reference indicators, it does not use the Delta method to compute
the corresponding standard error estimates).

(Note :

The observed independent variables (income, occup, and educ) in the
MIMIC model were correlated amongst themselves as identified in the
correlation matrix of the SEM program output:
1.000
.304 1.000
.305 .344 1.000

The structural equation indicated that the latent variable social had 26%
of its variance predicted (R 2 = .26), with 74% unexplained error variance
due random or systematic error, and variables not in the MIMIC model.
The T-values for the structural equation coefficients indicated that occup
(occupation) didn’t statistically significantly predict social (T = parameter
estimate divided by standard error = .097/.056 = 1.73 is less than t = 1.96 at
the .05 level of significance, two-tailed test), whereas income (T = 3.82) and
educ (T = 4.93) were statistically significant at the .05 level of significance.
The structural equation with coefficients, standard errors in parentheses
and associated T values are listed below.

Y102005.indb 296

3/22/10 3:26:34 PM

297

Multiple Indicator–Multiple Indicator Cause, Mixture

Structural Equation
Social = 0.23*income + 0.097*occup + 0.33*educ, Errorvar.= 0.74 , R² = 0.26
(0.061)
(0.056)
(0.068)
(0.17)
3.82
1.73
4.93
4.35
15.1.3 Model Modification
The original MIMIC model was therefore modified by dropping church
and occup. The MIMIC model diagram with these modifications now
appears in Figure 15.1b.
error
income

member

err_m

friends

err_f

social
educ
Figure 15.1b
Modified MIMIC Model.

The model modification fit criteria are more acceptable, indicating an
almost perfect fit of the data to the MIMIC model, since the Minimum Fit
Function c2 value was close to zero.
Goodness-of-Fit Statistics
Degrees of Freedom = 1
Minimum Fit Function Chi-Square = 0.19 (P = 0.66)
Root Mean Square Error of Approximation (RMSEA) = 0.0
90 Percent Confidence Interval for RMSEA = (0.0 ; 0.088)
P-Value for Test of Close Fit (RMSEA < 0.05) = 0.82
Normed Fit Index (NFI) = 1.00
Goodness-of-Fit Index (GFI) = 1.00

Measurement Equations
member = 0.63*social, Errorvar.= 0.60 , R² = 0.39
(0.08)
7.32
friends = 0.42*social, Errorvar. = 0.82 , R² = 0.17
(0.07) (0.06)
5.54 13.66

Y102005.indb 297

3/22/10 3:26:35 PM

298

A Beginner’s Guide to Structural Equation Modeling

( N ote : Because a matrix was used rather than raw data, standard errors
are not output for one of the reference indicator variables, member =
0.63*social. The HELP menu offers further explanation as noted above.)

Structural Equations
The structural equation now indicated two statistically significant predictor variables with R 2 = .36. This also implies that 64 percent of the latent
variable variance is left unexplained, mostly due to random or systematic
error or other variables not included in the MIMIC model.
social = 0.31*income + 0.42*educ, Errorvar. = 0.64 , R² = 0.36
(0.063) (0.064) (0.19)
5.01 6.65 3.39

MIMIC models permit the specification of one or more latent variables with
one or more observed variables as predictors of the latent variables. This
type of SEM model demonstrates how observed variables can be incorporated into theoretical models and tested. We followed the five basic steps in
SEM: model specification, model identification, model estimation, model
testing, and model modification to obtain our best model to data fit.

15.2 Mixture Models
Mixture models in SEM involve the analysis of observed variables that
are categorical and continuous. SEM was originally created using continuous variables in a sample variance–covariance matrix (Pearson correlation matrix with means and standard deviations); however, today
SEM models with nominal, ordinal, interval, and ratio-level observed
variables can be used in SEM. The use of a mixture of variables, however, requires using other types of matrices than the Pearson correlation matrix and associated variance–covariance matrix in SEM
programs. In the LISREL software program, PRELIS (Pre-LISREL) is
used to input, edit and handle raw data and produce the type of matrix
needed for the LISREL program (JÖreskog & Sorbom, 1996b). In PRELIS,
a variable is defined as continuous by the CO command (by default
the variable must have a minimum of 15 categories), the OR command
for ordinal variables, or the CL command for class or group variables.
PRELIS can output normal theory variance–covariance matrices (correlation between continuous variables), polychoric matrices (correlation
between ordered categorical variables), polyserial matrices (correlation

Y102005.indb 298

3/22/10 3:26:35 PM

Multiple Indicator–Multiple Indicator Cause, Mixture

299

between continuous and ordered categorical variable), and asymptotic
variance–covariance matrices (continuous and/or ordinal variables
with nonnormality), and augmented moment matrices (matrices with
variable means). Consequently, in LISREL, one would use PRELIS to create and save the appropriate variance–covariance matrix, conduct the
analysis as usual, and interpret the fit statistic using a robust model-fit
measure (Note: The sample variance–covariance matrix and asymptotic
covariance matrix with maximum likelihood estimation is required to
obtain the Satorra–Bentler robust c2 statistic).
15.2.1 Model Specification and Identification
The mixture model example uses variables from the SPSS data set bankloan.sav. This is a hypothetical data set that concerns a bank’s efforts to
reduce the rate of loan defaults. The file contains financial and demographic information on 850 past and prospective customers. The data set
is located in the SPSS Samples folder, our path location was:
C:\Program Files\SPSSInc\SPSS16\Samples\bankloan.sav
A theoretical model was hypothesized that financial Ability was a predictor of Debt. The observed variables age, level of education, years with
current employer, years at current address, and household income in thousands were used as indicators of the latent independent variable, Ability.
The observed variables credit card debt in thousands and other debt in
thousands were used as indicators of the latent dependent variable, Debt.
The SPSS save file (bankloan.sav) was imported and saved as a PRELIS
System File (bankloan.psf). The File, and then Import Data commands,
were used along with the Save As command noted in the following two
dialog boxes.

Y102005.indb 299

3/22/10 3:26:36 PM

300

A Beginner’s Guide to Structural Equation Modeling

We now opened the PRELIS System File, bankloan.psf, and deleted the
variable DEBTINC by clicking on the variable name using the right mouse
button. Next, we deleted the last four variables, Default, preddef1, preddef2, and preddef3, leaving seven variables for the theoretical model. We
decided that these five variables (DEBTINC, Default, preddef1, preddef2,
and preddef3) were not good indicators in our theoretical model. (Note:
The following dialog boxes will appear if you right mouse click on the
variable name).

Y102005.indb 300

3/22/10 3:26:36 PM

301

Multiple Indicator–Multiple Indicator Cause, Mixture

The level of education variable was ordinal (OR), while all other variables were considered continuous (CO). The mixture model for Ability predicting Debt is therefore represented in Figure 15.2a.

AGE

CREDDEBT

EMPLOY

Ability

Debt

OTHDEBT
ADDRESS

INCOME
Figure 15.2a
Mixture Model.

15.2.2 Model Estimation and Testing
In LISREL, we can now write a PRELIS program that will read in the
data and output a polyserial correlation matrix, bankloan.mat. (Note: The
PRELIS program only requires a few lines of code to read in the data
and output 8 different types of matrices [Jöreskog & Sörbom, 1996b,
p. 92–93]). The title of the program is Polyserial correlation matrix. The DA
command specifies seven input variables (NI = 7) with 850 observations
(NO = 850); missing data is identified by a zero (MI = 0) and treat missing
data listwise (TR = LI). The SY command identifies the PRELIS system
file (bankloan.psf). The OU command identifies the type of matrix to be
computed—that is, polyserial matrix (MA = PM)—and the name of the
polyserial matrix (PM = bankloan.mat). The PRELIS program was entered
and saved as bankloan.pr2. (Note: We click on the run-P icon to execute
PRELIS programs.)

Y102005.indb 301

3/22/10 3:26:37 PM

302

A Beginner’s Guide to Structural Equation Modeling

Polyserial correlation matrix
DA NI = 7 NO=850 MI = 0 TR = LI
SY FI = bankloan.psf
CO AGE
OR ED
CO EMPLOY
CO ADDRESS
CO INCOME
CO CREDDEBT
CO OTHDEBT
OU MA = PM PM = bankloan.mat

Two variables, EMPLOY and ADDRESS, had missing data leaving an
effective sample size of N = 723. The resulting saved polyserial correlation
matrix, bankloan.mat, is now used in our mixture model program analysis.
The LISREL–SIMPLIS program for the mixture model would be:
Mixture Model using Polyserial Correlation Matrix
Observed Variables: AGE ED EMPLOY ADDRESS INCOME CREDDEBT
OTHDEBT
Sample Size 723
Correlation Matrix
1.000
0.041 1.000
0.524 −0.163 1.000
0.589 0.099 0.335 1.000
0.454 0.251 0.610 0.299 1.000
0.261 0.138 0.380 0.150 0.559 1.000
0.320 0.162 0.411 0.166 0.598 0.647 1.000
Means: 35.903 0.000 9.593 9.216 49.732 1.665 3.271
Standard Deviations: 7.766 1.000 6.588 6.729 40.243 2.227
3.541
Latent Variables Ability Debt
Relationships
AGE ED EMPLOY ADDRESS INCOME = Ability
CREDDEBT OTHDEBT = Debt
Debt = Ability
Number of Decimals = 3
Path Diagram
End of Problem

The theoretical model analysis indicated that the normal theory model fit
results were not adequate (c2 = 428.22, df = 13, p = 0.0001, RMSEA = 0.210).
We therefore examined the modification indices to determine any substantive model modifications.

Y102005.indb 302

3/22/10 3:26:38 PM

303

Multiple Indicator–Multiple Indicator Cause, Mixture

0.64

AGE
0.60

0.97

0.49

0.82

0.16
0.71

EMPLOY

Ability

0.76

Debt

0.43

ADDRESS

CREDDEBT

0.41

OTHDEBT

0.30

0.77
0.84

0.85
Chi-Square = 428.22, df = 13, P = 0.0001, RMSEA = 0.210

0.27

INCOME

Figure 15.2b
Mixture Model output.

15.2.3 Model Modification
The modification indices for Figure 15.2b suggested the following:
The Modification Indices Suggest to Add an Error Covariance
Between
and
Decrease in Chi-Square
New Estimate
AGE
EMPLOY
EMPLOY
ADDRESS
ADDRESS
INCOME
INCOME
INCOME
INCOME
INCOME

CREDDEBT
AGE
ED
OTHDEBT
AGE
CREDDEBT
OTHDEBT
AGE
ED
ADDRESS

10.0
34.0
144.9
8.9
182.0
16.3
14.8
60.3
74.6
38.0

−1.30
8.11
−2.28
−1.76
20.71
8.41
12.86
−72.01
9.49
−45.95

We felt that EMPLOY (years with current employer), ED (education level),
ADDRESS (years at current address), and AGE were very much related to
each other. We therefore added the following commands in the LISREL–
SIMPLIS program to correlate their respective error covariance:
Let error covariance of EMPLOY and ED correlate
Let error covariance of ADDRESS and AGE correlate
Let error covariance of EMPLOY and AGE correlate

Y102005.indb 303

3/22/10 3:26:38 PM

304

A Beginner’s Guide to Structural Equation Modeling

Our results continued to indicate a poor model fit (Normal theory c2 =
47.73, df = 10, p = 0.0001, RMSEA = 0.072). We therefore examined additional modification indices from our second analysis:
The Modification Indices Suggest to Add an Error Covariance
Between
and
Decrease in Chi-Square
New Estimate
ED
AGE
12.5
−0.75
ADDRESS
ED
14.8
0.73
ADDRESS
EMPLOY
30.8
6.53

These modifications also seemed reasonable given how years with current
employer; years at current address, age, and education were related. We
therefore added the following additional command lines to the LISREL–
SIMPLIS program:
Let error covariance of EMPLOY and ADDRESS correlate
Let error covariance of ADDRESS and ED correlate
Let error covariance of AGE and ED correlate

The final theoretical model was therefore modified to include all of these
error covariance correlations with corresponding command lines added
to the LISREL–SIMPLIS program (JÖreskog & Sorbom, 1996c). The final
LISREL–SIMPLIS program, bankloan.psf, was therefore modified as follows:
Mixture Model Using Polyserial Correlation Matrix
Observed Variables: AGE ED EMPLOY ADDRESS INCOME CREDDEBT
OTHDEBT
Sample Size 723
Correlation Matrix
1.000
0.041 1.000
0.524 −0.163 1.000
0.589 0.099 0.335 1.000
0.454 0.251 0.610 0.299 1.000
0.261 0.138 0.380 0.150 0.559 1.000
0.320 0.162 0.411 0.166 0.598 0.647 1.000
Means: 35.903 0.000 9.593 9.216 49.732 1.665 3.271
Standard Deviations: 7.766 1.000 6.588 6.729 40.243 2.227
3.541
Latent Variables Ability Debt
Relationships
AGE ED EMPLOY ADDRESS INCOME = Ability
CREDDEBT OTHDEBT = Debt
Debt = Ability
Let error covariance of EMPLOY and ED correlate
Let error covariance of ADDRESS and AGE correlate
Let error covariance of EMPLOY and AGE correlate

Y102005.indb 304

3/22/10 3:26:38 PM

Multiple Indicator–Multiple Indicator Cause, Mixture

Let error covariance
Let error covariance
Let error covariance
Number of Decimals =
Path Diagram
End of Problem

305

of EMPLOY and ADDRESS correlate
of ADDRESS and ED correlate
of AGE and ED correlate
3

The theoretical model now had an adequate fit to the bank loan data
(Normal Theory c2 = 5.69, df = 7, p = 0.57607, RMSEA = 0.00). However,
we recalled that mixture models should report robust statistics which
will require using an asymptotic covariance matrix in addition to
the sample covariance matrix and maximum likelihood estimation
method. So we next describe how to obtain the Satorra–Bentler scaled
robust statistic.
15.2.4 Robust Statistic
Our SEM analysis required a polyserial correlation matrix because we
had a mixture of variables (ordinal and continuous). We should therefore
be reporting a robust chi-square statistic not a normal theory chi-square
statistic. How do I obtain the Satorra–Bentler Chi-square robust statistic
value? We first open the PRELIS system file, bankloan.psf, and then save
a covariance matrix (bankloan.cov) and an asymptotic covariance matrix
(bankloan.acm) using the Statistics pull down menu and Output Option as
seen in the dialog box below:

Y102005.indb 305

3/22/10 3:26:39 PM

306

A Beginner’s Guide to Structural Equation Modeling

Our LISREL–SIMPLIS program is now modified to include the Covariance
matrix from file, Asymptotic Covariance Matrix from File, and Method of
Estimation: Maximum Likelihood commands. The computer output under
Goodness-of-Fit statistics will now include the robust Satorra–Bentler
scaled chi-square statistic. The updated LISREL–SIMPLIS program with
these commands would be:
Mixture Model using Polyserial Correlation Matrix
Observed Variables: AGE ED EMPLOY ADDRESS INCOME CREDDEBT
OTHDEBT
Sample Size 723
Covariance matrix from file bankloan.cov
Asymptotic Covariance Matrix from File bankloan.acm
Method of Estimation: Maximum Likelihood
Latent Variables Ability Debt
Relationships
AGE ED EMPLOY ADDRESS INCOME = Ability
CREDDEBT OTHDEBT = Debt
Debt = Ability
Let error covariance of EMPLOY and ED correlate
Let error covariance of ADDRESS and AGE correlate
Let error covariance of EMPLOY and AGE correlate
Let error covariance of EMPLOY and ADDRESS correlate
Let error covariance of ADDRESS and ED correlate
Let error covariance of AGE and ED correlate
Number of Decimals = 3
Path Diagram
End of Problem

The final theoretical model with the Satorra–Bentler scaled chi-square statistic reported is shown in Figure 15.2c. The Satorra–Bentler Scaled c2 =
3.419, df = 7, p = 0.844 for the theoretical model compared to the Normal
Theory c2 = 5.69, df = 7, p = 0.57607. We should expect the robust statistic to
indicate a better model fit.
The SEM mixture model permits continuous and categorical variables
to be used in a theoretical model. The mixture model however uses a different correlation matrix than the traditional Pearson correlation matrix
with means and standard deviations. Consequently, you will need to use
PRELIS to read in a data set and output a polyserial correlation matrix.
Additionally, you will need to save a covariance matrix and an asymptotic covariance matrix in PRELIS and include it in the SIMPLIS program along with maximum likelihood estimation method to obtain the
Satorra–Bentler scaled chi-square statistic for appropriate interpretation
of the mixture model.

Y102005.indb 306

3/22/10 3:26:39 PM

307

Multiple Indicator–Multiple Indicator Cause, Mixture

0.75

AGE

–0.11
0.23

0.50
ED

0.94

0.24

–0.33

0.44

–0.02

0.57

0.65

EMPLOY

0.90

ADDRESS

0.08

INCOME

Ability

0.75

Debt

0.32

0.14

CREDDEBT

0.41

OTHDEBT

0.29

0.77
0.84

0.96

Satorra-Bentler Scaled Chi-Square = 3.42, df = 7, P = 0.84377

Figure 15.2c
Modified Mixture Model (Satorra–Bentler scaled Chi-square).

15.3 Multilevel Models
Multilevel models in SEM are so named because of the hierarchical nature
of data in a nested research design. For example, in education a student’s
academic achievement is based in classrooms, so students are nested in
classrooms, teachers are nested within schools, and schools are nested
within districts. The nested research design is in contrast to a crossed
research design where every level is represented. In multilevel models our
interest is in the effects at different levels given the clustered nature of the
data. A simple schematic will illustrate multilevel versus crossed designs.
Multilevel Design: Four teachers are indicated at two schools; however,
teachers 1 and 2 are in School A, while teachers 3 and 4 are in School B.
School
Teacher

A
1

B
2

Crossed Design: Four teachers are indicated at two schools with all four
teachers in both schools.
School
Teacher

A
1234

B
1234

Several textbooks introduce and present excellent multilevel examples, so
we refer you to those for more information on the analysis of multilevel
models in SEM (Heck & Thomas, 2000; Hox, 2002). We have also provided
a few journal article references that have used the multilevel approach.

Y102005.indb 307

3/22/10 3:26:39 PM

308

A Beginner’s Guide to Structural Equation Modeling

LISREL provides an extensive HELP library on multilevel modeling that
includes an overview of multilevel modeling; differences between OLS
and multilevel random coefficient models (MRCM); latent growth curve
models; testing of contrasts; analysis of two-level repeated measures data;
multivariate analysis of educational data; multilevel models for categorical
response variables; and examples using air traffic control data, school, and
survey data. Consequently, you are encouraged to use the HELP library in
LISREL for more information and examples using PRELIS and SIMPLIS
or read about the new statistical features in LISREL by Karl Jöreskog, Dag
Sörbom, Stephen du Toit and Mathilda du Toit (2001).
In LISREL, you will be using the multilevel tool bar menu to demonstrate variance decomposition, which is a basic multilevel model (equivalent
to a one-way ANOVA with random effects). The multilevel null model is a
preliminary first step in a multilevel analysis because it provides important information about the variability of the dependent variable. You
should always create a null model (intercept only) to serve as a baseline
for comparing additional multilevel models when you add variables to
test whether they significantly reduce the unexplained variability in the
dependent variable (response or outcome variable).
In LISREL 8.8, student version, find the directory that is labeled, LISREL
8.8 Student Examples, then select the mlevelex folder, next select the files of type
which indicates PRELIS DATA (*.psf). You will now see PRELIS SYSTEM
FILES (*.psf). Select MOUSE. The dialog box should look like the following:

MOUSE.PSF is a nested data set with nine weight measurements taken
at nine time periods on 82 mice. The data set should contain n = 738 rows
of data (9 × 82), however, the data set only contains n = 698 rows of data
because some weights are missing for the mice, for example, iden2 = 43,
44, 45, etc. The variables in the MOUSE.PSF system file are iden2, iden1,
weight, constant, time, timesq, and gender. The dialog box below displays
the spreadsheet with these variables.

Y102005.indb 308

3/22/10 3:26:40 PM

Multiple Indicator–Multiple Indicator Cause, Mixture

309

The multilevel command now appears on the tool bar menu with linear
and nonlinear model options. Now select Linear Model, and then Title and
Options. You will be specifying variables for each of the options shown
here, but this is accomplished by selecting NEXT after you enter the information for Title and Options. You can enter the title Mouse Data: Variance
Decomposition in the dialog box as indicated below, and then click NEXT.

Y102005.indb 309

3/22/10 3:26:40 PM

310

A Beginner’s Guide to Structural Equation Modeling

This takes us to the Identification Variables dialog box where you will
add ident2 to level 3 and ident1 to level 2. The variable ident2 ranges from
1 to 82 and identifies the unique mouse, while ident1 indicates the 9 time
measurements and ranges from 1 to 9. The dialog box should look like the
one below:

You again click NEXT. This takes us to the Select Response and
Fixed Variables dialog box where you add weight as the select response
(dependent variable and constant as a fixed effect to create an intercept only
(null) model). Be sure to unselect the Intercept box in this dialog box as
indicated below:

Y102005.indb 310

3/22/10 3:26:41 PM

Multiple Indicator–Multiple Indicator Cause, Mixture

311

You again click NEXT. This takes you to the Random Variables dialog
box where you will add constant to both random Level 3 and random Level
2. Constant is the intercept term for the response variable (weight) and
associates an error term for the Level 3 and Level 2 equations. Be sure to
unselect the Intercept boxes in this dialog box for ALL RANDOM LEVELS
as indicated below:

Y102005.indb 311

3/22/10 3:26:41 PM

312

A Beginner’s Guide to Structural Equation Modeling

Now click FINISH and a PRELIS program, mouse.pr2 is written.

Y102005.indb 312

3/22/10 3:26:42 PM

Multiple Indicator–Multiple Indicator Cause, Mixture

313

The PRELIS program is executed by clicking the run P (Run PRELIS)
on the tool bar menu. The PRELIS computer output will now indicate the
fixed and random results for the baseline model (intercept only). (Note:
Do not use the term constant in your model and also select intercept in the
dialog boxes.)
15.3.1 Constant Effects
The PRELIS computer output for the baseline model (constant) is as follows:
+-------------------------------------+
| FIXED PART OF MODEL |
+-------------------------------------+
-----------------------------------------------------------COEFFICIENTS
BETA-HAT
STD.ERR.
Z-VALUE
PR > |Z|
-----------------------------------------------------------constant
28.63410
0.57021
50.21634
0.00000
+--------------------------------+
| -2 LOG-LIKELIHOOD |
+--------------------------------+
DEVIANCE= -2*LOG(LIKELIHOOD) = 5425.490015929897
NUMBER OF FREE PARAMETERS = 3
+---------------------------------------+
| RANDOM PART OF MODEL |
+---------------------------------------+
-----------------------------------------------------------LEVEL 3
TAU-HAT
STD.ERR. Z-VALUE
PR > |Z|
-----------------------------------------------------------constant/constant 11.32910 4.25185
2.66451
0.00771
-----------------------------------------------------------LEVEL 2
TAU-HAT
STD.ERR. Z-VALUE
PR > |Z|
-----------------------------------------------------------constant/constant 130.32083 7.42514 17.55130
0.00000

15.3.2 Time Effects
The second multilevel analysis includes adding time to the fixed variable
list. To do so, click on Multilevel, Linear Models, and then Select Response
and Fixed Variables in the drop-down menu. Now add time to the fixed
variable list as indicated in the dialog box below. You will click NEXT.
Do not change the Random Variables dialog box that appears; simply click
FINISH.

Y102005.indb 313

3/22/10 3:26:42 PM

314

A Beginner’s Guide to Structural Equation Modeling

The following PRELIS program will appear in a dialog box with time
added to the FIXED command. To run the updated PRELIS file, mouse.pr2,
click on run P (Run PRELIS).

Y102005.indb 314

3/22/10 3:26:43 PM

Multiple Indicator–Multiple Indicator Cause, Mixture

315

The PRELIS computer output for both constant (baseline) plus time is
as follows:
+-------------------------------------+
| FIXED PART OF MODEL |
+-----------------------------------+
-----------------------------------------------------------COEFFICIENTS BETA-HAT STD.ERR. Z-VALUE PR > |Z|
-----------------------------------------------------------constant 9.09586 0.60387 15.06258 0.00000
time 4.09218 0.06258 65.39108 0.00000
+--------------------------------+
| -2 LOG-LIKELIHOOD |
+--------------------------------+
DEVIANCE= -2*LOG(LIKELIHOOD) = 4137.578760208256
NUMBER OF FREE PARAMETERS = 4
+--------------------------------------+
| RANDOM PART OF MODEL |
+---------------------------------------+
-----------------------------------------------------------LEVEL 3
TAU-HAT
STD.ERR.
Z-VALUE
PR > |Z|
-----------------------------------------------------------constant/constant
20.69397 3.53655
5.85146
0.00000
-----------------------------------------------------------LEVEL 2
TAU-HAT
STD.ERR.
Z-VALUE
PR > |Z|
-----------------------------------------------------------constant/constant
16.46288 0.93806
17.54996 0.00000

15.3.3 Gender Effects
We repeat this process a third time to add gender to the fixed variables for a
final multilevel analysis. To do so, click on Multilevel, Linear Models, and
then Select Response and Fixed Variables in the drop-down menu. Now
add gender to the fixed variable list as indicated in the dialog box below.
You will click NEXT. Do not change the Random Variables dialog box that
appears; simply click FINISH. The Select Response and Fixed Variables dialog box should look like the following:

Y102005.indb 315

3/22/10 3:26:43 PM

316

A Beginner’s Guide to Structural Equation Modeling

The following PRELIS program will appear in a dialog box with gender
added to the FIXED command. To run the updated PRELIS file, mouse.pr2,
click on run P (Run PRELIS). The following PRELIS program should appear:

Y102005.indb 316

3/22/10 3:26:43 PM

Multiple Indicator–Multiple Indicator Cause, Mixture

317

The PRELIS computer output with constant, time, and gender is as
follows:
+-----------------------------------+
| FIXED PART OF MODEL |
+-----------------------------------+
-----------------------------------------------------------COEFFICIENTS BETA-HAT
STD.ERR.
Z-VALUE
PR >
|Z|
-----------------------------------------------------------constant
9.07800
0.58325
15.56442
0.00000
time
4.08714
0.06261
65.28249
0.00000
gender 1.42015
0.50199
2.82904
0.00467
+--------------------------------+
| -2 LOG-LIKELIHOOD |
+--------------------------------+
DEVIANCE = -2*LOG(LIKELIHOOD) = 4129.941071012016
NUMBER OF FREE PARAMETERS = 5
+---------------------------------------+
| RANDOM PART OF MODEL |
+---------------------------------------+
-----------------------------------------------------------LEVEL 3
TAU-HAT
STD.ERR.
Z-VALUE
PR > |Z|
-----------------------------------------------------------constant/constant 18.68475
3.22290
5.79750 0.00000
-----------------------------------------------------------LEVEL 2
TAU-HAT
STD.ERR.
Z-VALUE PR > |Z|
-----------------------------------------------------------constant/constant 16.46249
0.93804
17.54996 0.00000

You have now created and run three different PRELIS programs to obtain
the multilevel analysis results for an intercept model (model 1), intercept and time model (model 2), and an intercept, time, and gender model
(model 3). The PRELIS program, mouse.pr2, was updated each time you
changed the number of fixed variables. The three PRELIS programs are
listed below where it is easily seen that the FIXED command changed as
you added additional hypothesized variables to obtain a better prediction
of the unexplained variability of the response variable (weight).

Y102005.indb 317

3/22/10 3:26:44 PM

318

A Beginner’s Guide to Structural Equation Modeling

Model 1—Intercept Only
OPTIONS OLS=YES CONVERGE=0.001000 MAXITER=10 OUTPUT=STANDARD ;
TITLE=Mouse Data: Variance Decomposition;
SY=’C:\LISREL 8.8 Student Examples\MLEVELEX\MOUSE.PSF’;
ID3=iden2;
ID2=iden1;
RESPONSE=weight;
FIXED=constant;
RANDOM2=constant;
RANDOM3=constant;
Model 2 – Intercept + Time
OPTIONS OLS=YES CONVERGE=0.001000 MAXITER=10 OUTPUT=STANDARD ;
TITLE=Mouse Data: Variance Decomposition;
SY=’C:\LISREL 8.8 Student Examples\MLEVELEX\MOUSE.PSF’;
ID3=iden2;
ID2=iden1;
RESPONSE=weight;
FIXED=constant time;
RANDOM2=constant;
RANDOM3=constant;
Model 3 – Intercept + Time + Gender
OPTIONS OLS=YES CONVERGE=0.001000 MAXITER=10 OUTPUT=STANDARD ;
TITLE=Mouse Data: Variance Decomposition;
SY=’C:\LISREL 8.8 Student Examples\MLEVELEX\MOUSE.PSF’;
ID3=iden2;
ID2=iden1;
RESPONSE=weight;
FIXED=constant time gender;
RANDOM2=constant;
RANDOM3=constant;

The final multilevel equation is specified as:
Yij = b1 + b 2 Timeij + b 2 Genderij + uij + eij.
The PRELIS computer results are summarized in Table 15.1 for comparative purposes. (Note: Other multilevel models could include random
effects rather than only fixed effects.)
15.3.4 Multilevel Model Interpretation
The computer output for the three PRELIS multilevel programs are summarized in Table 15.1 for the variance decomposition of the response variable, weight. Model 1 provides a baseline model to determine if additional
variables help in reducing the amount of variability in weight. Model 2
with time added, substantially reduced the unexplained variability in

Y102005.indb 318

3/22/10 3:26:44 PM

319

Multiple Indicator–Multiple Indicator Cause, Mixture

Table 15.1
Summary Results for Multilevel Analysis of Mouse Weight
Multilevel Model
Fixed Factors

Model 1
Constant Only

Model 2
Constant + Time

Model 3 Constant +
Time + Gender

Intercept Only(B0)
Time (B1)
Gender (B2)

28.63 (.57)

9.09 (.60)
4.09 (.06)

9.07 (.58)
4.08 (.06)
1.42 (.50)

Level 2 error variance (eij)
Level 3 error variance (uij)
ICC
Deviance (-2LL)
Df
Chi-square
Difference
(df = 1)

130.32
11.33
.079 (8%)
5425.49
3

16.46
20.69
.556 (56%)
4137.57
4

16.46
18.68
.532 (53%)
4129.94
5

1287.92

7.63

Note: c2 = 3.84, df = 1, p = .05.

weight (c2 = 1287.92, df = 1). Model 3 with gender added also significantly
reduced the amount of unexplained variability in weight (c2 = 7.63, df = 1).
Therefore, mouse weight variability is statistically significantly explained
by time and gender fixed variables.
15.3.5 Intraclass Correlation
The intraclass correlation coefficient measures the relative homogeneity
within groups in ratio to the total variation. In ANOVA it is computed
as (Between-groups MS − Within-groups MS)/(Between-groups MS +
(n − 1)* Within-Groups MS), where n is the average number of cases in
each category of the independent variable. SPSS has a drop-down menu
option for computing the intraclass correlation coefficient in your data.
If the intraclass correlation coefficient is large and positive, then there
is no variation within the groups, but group means differ. It will be at
its largest negative value when group means are the same but there is
great variation within groups. Its maximum value is 1.0, but its maximum
negative value is (−1/(n − 1)). A negative intraclass correlation coefficient
occurs when between-group variation is less than within-group variation,
indicating a third variable is present with nonrandom effects on the different groups.
The presence of a significant intraclass correlation coefficient indicates the
need to employ multilevel modeling rather than OLS regression. The main
difference is in the standard errors of the parameters, which have smaller

Y102005.indb 319

3/22/10 3:26:44 PM

320

A Beginner’s Guide to Structural Equation Modeling

estimates in regression analysis if the intraclass correlation coefficient is
statistically significant, which can inflate (bias) the regression weights. The
intraclass correlation coefficient, using our results, is computed in SEM as:
ICC =
=

Φ3
Tau − Hat(Level − 3)
=
Φ 3 + Φ 2 Tau − Hat(Level − 3) + Tau − Hat(Level − 2)
11.33
= .079
11.33 + 130.32

Therefore, 8% of the variance in weight is explained in the baseline model.
It jumps dramatically when adding time as an explanatory variable to 56%
variance in weight, explained as a function of time. It drops modestly to
53% when adding gender to the equation. The 3% difference is not enough
to infer a nonsignificant effect; therefore time and gender significantly
explain 53% of the variance in mice weight.
15.3.6 Deviance Statistic
The deviance statistic is computed as −2lnL (likelihood function), which
is used to test for statistical difference in models between Model 1 (constant), Model 2 (constant + time), and Model 3 (constant + time + gender). I
chose the chi-square value of 3.84, df = 1, at the p = .05 level of significance
to test whether additional variables in the equation explained variance in
mice weight. The baseline deviance value was 5425.49. The chi-square difference test between this baseline deviance statistic and the second equation deviance value with time (–2lnL = 4137.57) indicated a difference of
1287.92, which is statistically significantly different than the tabled critical
chi-square value of 3.84. Consequently, time was a significant predictor
variable of mice weight. The model with time and gender indicated a deviance statistic of 4129.94 and had a difference from the previous deviance
statistic of 7.63, which was also statistically significantly different from
the critical tabled chi-square of 3.84. Consequently, time and gender were
statistically significant predictor variables of mice weight.

15.4 Summary
In this chapter, we have described MIMIC, mixture, and multilevel modeling, to further demonstrate the versatility of structural equation modeling. The first application presented a SEM model that had multiple
indicators of a latent variable where the latent variable was predicted
by multiple observed variables. We refer to this type of SEM model as a

Y102005_C015.indd 320

4/3/10 4:21:44 PM

Multiple Indicator–Multiple Indicator Cause, Mixture

321

Multiple Indicator and Multiple Cause (MIMIC) model. The next application involved models that used ordinal and continuous variables.
We refer to this type of SEM model as a Mixture Model. In this application, we learned that normal theory fit indices apply to continuous
variables that use a Pearson Correlation Matrix with means and standard deviations of the variables, but that other matrices should be used
when ordinal and continuous variables are present in the SEM model
(for example, polychoric or polyserial matrices). Our final application
involved analyzing nested data, which has become increasingly popular
in repeated measures, survey, and education data analysis because of
the hierarchical research design. In SEM, we refer to this type of model
as a Multilevel model, but in the research literature this type of model
is referred to by many different names—for example, hierarchical linear,
random-coefficient, variance-component modeling, or HLM.
The chapter presented only one example for each of the applications
because a more in depth coverage is beyond the scope of this book.
However, the LISREL software HELP library and examples can be searched
by using keywords to find other software examples and explanations. The
LISREL User Guide is also an excellent reference for other examples of
these applications. We now turn our attention to the next chapter where
other SEM applications are presented and discussed.
Exercises
1. Multiple Indicator–Multiple Cause
(MIMIC) Model
Create and run a LISREL–SIMPLIS program given the MIMIC model
below. Please interpret the results including any model modification,
significance of coefficients, and R2 value. The data set information is:
Observed Variables peer self income shift age
Sample Size 530
Correlation Matrix
1.00
.42 1.00
.24 .35 1.00
.13 .37 .25 1.00
.33 .51 .66 .20 1.00

The following MIMIC Model (next page) includes the latent variable job
satisfaction (satisfac), which is defined in Figure 15.3 by two observed variables: peer ratings and self ratings. A person’s income level, which shift
they work, and age are observed predictor variables of job satisfaction.
2. Mixture model
Given the following Miture Model in Figure 15.4 and data set information, write a LISREL program to test the Mixture Model. (Note: Robust

Y102005.indb 321

3/22/10 3:26:45 PM

322

A Beginner’s Guide to Structural Equation Modeling

Income

Peer Rating
Job
Satisfaction

Shift

Self Rating

Age

Figure 15.3
Job Satisfaction MIMIC model.

statistics require the raw data file, so no Satorra–Bentler scaled chisquare possible). The Mixture Model has six observed variables (Age,
Gender, Degree, Region, Hours, and Income) that define two latent variables (Person and Earning). A polyserial correlation matrix was created
where CO indicates continuous variable and OR indicates a categorical variable. Age (CO), Gender (OR), and Degree (OR) define Personal
characteristics, an independent latent variable (Person). Region (OR),
Hours (CO), and Income (CO) define dependent latent variable Earning
Power (Earning). Personal Characteristics (Person) is hypothesized to
predict Earning Power (Earning).
The data for the Mixture Model is:
Observed Variables: Age Gender Degree Region Hours Income
Correlation Matrix
1.000
0.487 1.000
0.236 0.206 1.000
0.242 0.179 0.253 1.000
0.163 0.090 0.125 0.481 1.000
0.064 0.040 0.025 0.106 0.136 1.000
Means 15.00 10.000 10.000 10.000 7.000 10.000
Standard Deviations 10.615 10.000 8.000 10.000 15.701
10.000
Sample Size 600

The Mixture Model diagram is:
Age

Gender

Degree

Region

Person

Earning

Hours

Income

Figure 15.4
Earning Power Mixture Model.

Y102005.indb 322

3/22/10 3:26:45 PM

Multiple Indicator–Multiple Indicator Cause, Mixture

323

3. Multilevel Model
You will need to access the directory, LISREL 8.8 Student Examples. Click
on the mlevelex folder and select the PRELIS system file, income.psf, which
contains the variables region, state, age, gender, marital, etc. There are nine
regions with 51 states nested within the regions. The sample size is n =
6062. It is hypothesized that income varies by state within region.
Open the PRELIS system file, income.psf, and run three PRELIS multilevel model programs. The first model will be an intercept only model
with income as the response variable, Level 3 or ID3 = region, and Level
2 or ID2 = state. The second PRELIS program will add gender as a fixed
variable. The third PRELIS program will add an additional variable,
marital, as a fixed variable. Use the multilevel pull-down menu on the
tool bar to create the programs. (Note: Unselect the Intercept box in
each dialog box).
List Model 1, Model 2, and Model 3 PRELIS programs and summarize the output from the three PRELIS programs in a table. You
will need to hand calculate the intraclass correlation coefficient and
be sure to interpret the comparative results in the table. The MODEL
1 dialog box should look like the following:

Y102005.indb 323

3/22/10 3:26:46 PM

324

A Beginner’s Guide to Structural Equation Modeling

Suggested Readings
Multiple Indicator–Multiple Cause Models
Anderson, K. G., Smith, G. T., & McCarthy, D. M. (2005). Elementary school drinking: The role of temperament and learning. Psychology of Addictive Behaviors,
19(1), 21–27.
Sanchez-Perez, M., & Iniesta-Bonillo, M. A. (2004, Winter). Consumers felt commitment towards retailers: Index development and validation. Journal of
Business and Psychology, 19(2), 141–159.
Shenzad, S. (2006). The determinants of child health in Pakistan: An economic
analysis. Social Indicators Research, 78, 531–556.

Y102005.indb 324

3/22/10 3:26:46 PM

Multiple Indicator–Multiple Indicator Cause, Mixture

325

Mixture Models
Bagley, M. N., & Mokhtarian, P. L. (2002). The impact of residential neighborhood
type on travel behavior: A structural equations modeling approach. The
Annals of Regional Science, 36, 279-297.
Loken, E. (2004). Using latent class analysis to model temperament types.
Multivariate Behavioral Research, 39(4), 625-652.
Lubke, G. H., & Muthen, B. (2005). Investigating population heterogeneity with
factor mixture models. Psychological Methods, 10, 21–39.

Multilevel Models
Bryan, A., Schmiege, S. J., & Broaddus, M. R. (2007). Mediational analysis in HIV/
AIDS research: Estimating multivariate path analytic models in a structural
equation modeling framework. AIDS Behavior, 11, 365–383.
Everson, H. T., & Millsap, R. E. (2004). Beyond individual differences: Exploring
school effects on SAT scores. Educational Psychologist, 39(3), 157–172.
Trautwein, U., Ludtke, O., Schnyder, I., & Niggli, A. (2006). Predicting homework
effect: Support for a domain-specific, multilevel homework model. Journal of
Educational Psychology, 98, 438–456.

References
Heck, R. H., & Thomas, S. L. (2000). An introduction to multilevel modeling techniques.
Mahwah, NJ: Lawrence Erlbaum.
Hox, J. (2002). Multilevel analysis: Techniques and applications. Mahwah, NJ:
Lawrence Erlbaum.
Jöreskog, K., & Sörbom, D. (1996a). LISREL 8: User’s reference guide. Chicago, IL:
Scientific Software International.
Jöreskog, K., & Sörbom, D. (1996b). PRELIS2: User’s reference guide. Chicago, IL:
Scientific Software International.
Jöreskog, K., & Sörbom, D. (1996c). LISREL 8: Structural equation modeling with the
SIMPLIS command language. Chicago, IL: Scientific Software International.
Jöreskog, K., Sörbom, D., du Toit, S., & du Toit, M. (2001). LISREL8: New statistical
features. Chicago, IL: Scientific Software International.

Y102005.indb 325

3/22/10 3:26:46 PM

16
Interaction, Latent Growth,
and Monte Carlo Methods

Key Concepts
Main effects and Interaction Effects
Types of Interaction Effects: continuous nonlinear, categorical,
latent variable
Longitudinal data analysis using growth curve models
Monte Carlo methods

16.1 Interaction Models
Most SEM models have assumed that the relations in the models were
linear (i.e., the relations among all variables, observed and latent, are represented by linear equations). Several studies have been published where
nonlinear and interaction effects are used in multiple regression models;
however, these effects have seldom been tested in path models, and you
will infrequently find nonlinear factor models. It should not be surprising
to find that for several decades structural equation modeling has been
based on Linear Structural Relations (LISREL).
SEM models with nonlinear and interaction effects are now possible and can easily be modeled with recent versions of SEM software.
However, there are several types of nonlinear and interaction effects:
categorical, product indicant, nonlinear, two-stage least squares, and
latent variable using normal scores. For continuous observed variables, a
nonlinear relationship could exist between two observed variables (i.e.,
X1 and X 2 are curvilinear); a quadratic (nonlinear) term in the model
(i.e., X 2 = X 21) ; or a product of two observed variables (e.g., X3 = X1X 2).
327

Y102005.indb 327

3/22/10 3:26:47 PM

328

A Beginner’s Guide to Structural Equation Modeling

These three different types of interaction effects all involve continuous
observed variables. For categorical observed variables, interaction effects
are similar to analysis-of-variance and use the multiple-group SEM
model (Schumacker & Rigdon, 1995). These continuous variable and categorical variable approaches also apply to latent variables (e.g., latent
variable and latent class).
Given that so many different approaches exist, the categorical, latent variable and two-stage least squares examples will be illustrated. Categorical
interaction uses a multigroup (multisample) SEM model. The latent variable interaction uses the product of individual latent variable scores that
are computed and added to the PRELIS system file.
16.1.1 Categorical Variable Approach
In the categorical variable interaction approach, different groups (samples) are defined by the different levels of the interaction variable. The
basic logic is that if interaction effects are present, then certain parameters should have different values in different groups (samples). Both main
effects and interaction effects can be determined by using different groups
(samples) to test for differences between intercepts and slopes. You accomplish this by running two different SEM categorical variable interaction
models: (1) main effects for group differences holding slopes constant, and
(2) interaction effects for group differences with both intercepts and slopes
estimated. These models are sometimes referred to as intercept only and
intercept-slope models.
The following two LISREL–SIMPLIS programs analyze data for two
groups: boys versus girls, where group represents the categorical variable.
Separate covariance matrices and means on the dependent and independent variable are input to estimate the prediction of a math score, given a
pretest score. The means are required; otherwise, the intercept values will
be zero. The first LISREL–SIMPLIS program includes Equation: Math =
CONST Pretest for the girls, but only Equation: Math = CONST for the
boys, which permits different intercept values to be estimated while keeping the slopes equal in the two groups:
Group Girls: Math and Pretest Scores
Observed Variables: Math Pretest
Covariance Matrix:
181.349
84.219 182.821
Means: 82.15 78.35
Sample Size: 373
Equation: Math = CONST Pretest
Group Boys: Math and Pretest Scores

Y102005.indb 328

3/22/10 3:26:47 PM

Interaction, Latent Growth, and Monte Carlo Methods

329

Covariance Matrix:
174.485
34.468 161.869
Means: 48.75 46.98
Sample Size: 249
Equation: Math = CONST
End of Problem

The results indicated that the slopes were equal (slope = .37), and the
intercepts were different (53.26 versus 31.43). The main effect model for
differences in intercepts with equal slopes, however, was not an acceptable fit (c2 = 12.24, p = .002, df = 2).
Girls Group:
Math = 53.26 + 0.37*Pretest, Errorvar. = 155.07, R² = 0.14
(3.04) (0.038)
(8.81)
17.53
9.73
17.59

Boys Group:
Math = 31.43 + 0.37*Pretest, Errorvar. = 155.07, R² = 0.12
(1.95)
(0.038)
(8.81)
16.13
9.73
17.59

The second LISREL–SIMPLIS program uses the Equation: Math =
CONST Pretest in both groups, thus specifying that both intercepts
and slopes are being tested for group differences. Conceptually, this
implies a difference in the means (intercept) and a difference in the rate
of change (slope).
Group Girls: Math and Pretest Scores
Observed Variables: Math Pretest
Covariance Matrix:
181.349
84.219 182.821
Means: 82.15 78.35
Sample Size: 373
Equation: Math = CONST Pretest
Group Boys: Math and Pretest Scores
Covariance Matrix:
174.485
34.468 161.869
Means: 48.75 46.98
Sample Size: 249
Equation: Math = CONST Pretest
End of Problem

Y102005.indb 329

3/22/10 3:26:47 PM

330

A Beginner’s Guide to Structural Equation Modeling

The results indicated that the intercepts (46.06 versus 38.75) and slopes
(.46 versus .21) were different in the two groups. This model with main
and interaction effects present had an acceptable model fit (c2 = 1.98, p =.16,
df = 1). The main effect for group differences in math exam scores is given
by the difference in the CONST values: 46.06 − 38.75 = 7.31. The interaction
effect is given by the difference in the slope estimates of pretest values for
the two groups: .46 − .21 = .25.
Girls group:
Math = 46.06 + 0.46*Pretest, Errorvar. = 154.85, R2 = 0.20
(3.80) (0.048)
(8.80)
12.13
9.65
17.59

Boys group:
Math = 38.75 + 0.21*Pretest, Errorvar.= 154.85, R2 = 0.045
(3.03)
(0.062)
(8.80)
12.81
3.43
17.59

A categorical variable interaction model can represent a wide variety of
interaction effects, including higher-order interactions, without requiring
any substantial new methodological developments. This approach can
also be used regardless of whether the interaction intensifies or mutes the
effects of the individual variables. Because the interaction effect is represented in the difference between groups (samples), the researcher is able to
test linear relations of variables within each group (sample), thus avoiding
any potential complications in fitting the model. Finally, multiple group
(sample) programs permit parameter constraints across groups thereby
permitting many different hypotheses of group differences.
The categorical interaction approach, however, does have certain weaknesses (e.g., smaller subsamples of the total sample size are used). This
could be a serious problem if some groups have low sample sizes that
affect group parameter estimates. This reduction in sample size could
also affect the results of the c2 difference tests. Thus, it is possible that the
categorical-variable approach may yield group samples that are too small,
resulting in a c2 test statistic that misleads the researcher into believing
that an interaction effect exists, whether it does or not. A possible solution is to minimize the number of distinct parameters being compared in
the model by fixing certain parameters to be invariant across the samples
being compared.
The categorical-variable interaction approach is not recommended when
hypothesizing interaction using continuous variables. The basic logic is
that there is a loss of information when reducing a continuous variable
to a categorical variable, for purposes of defining a group (i.e., recode age

Y102005.indb 330

3/22/10 3:26:47 PM

331

Interaction, Latent Growth, and Monte Carlo Methods

into young and old categories). Group misspecification can also occur
when forming groups. Where does one choose the point for dividing a
continuous variable into a categorical variable to form the groups? How
do you justify the arbitrary cut value (i.e., mean, median, or quartile)?
Random-sampling error also ensures that some cases would be misclassified, violating some basic assumptions about subject membership in a
particular group.
16.1.2 Latent Variable Interaction Model
A latent variable interaction model would hypothesize that the independent latent variables (ksi1 and ksi2), as well as the product of ksi1 and ksi2
(ksi12), predict a dependent latent variable (eta). The latent variable interaction model is diagrammed in Figure 16.1a.

KSI1
V3

gamma1
KSI12

gamma3

gamma2

ETA

V2
V1

KSI2

Figure 16.1a
Latent Variable Interaction Model (Schumacker, 2002).

16.1.2.1 Computing Latent Variable Scores
The latent interaction variable approach uses a PRELIS system file and
intermediate steps to create and put latent variable scores into the PRELIS
system file (eta, ksi1, ksi2, and ksi12).
In LISREL, click on File, Open, and then locate the PRELIS SYSTEM
FILE, raw.psf.

Y102005.indb 331

3/22/10 3:26:48 PM

332

A Beginner’s Guide to Structural Equation Modeling

You should see the following PRELIS system file spreadsheet with
the 9 variables (V1 –V9). Also, a tool bar menu will appear across the
top.

You can create a dependent latent variable score and two independent latent variable scores, which will automatically be added to the
PRELIS SYSTEM FILE, raw.psf, by using the following LISREL–SIMPLIS
program:
Computing Latent Variable Scores
Observed Variables V1-V9
Raw Data from File raw.psf
Latent Variables : eta ksi1 ksi2
Relationships:
V1 = 1*eta
V2-V3 = eta
V4 = 1*ksi1
V5-V6 = ksi1
V7 = 1*ksi2
V8-V9 = ksi2
PSFfile raw.psf
End of Problem

You will need to close then open the PRELIS system file, raw.psf,
before you will see values for the three latent variables: eta, ksi1, and ksi2.)

( Note :

Y102005.indb 332

3/22/10 3:26:48 PM

Interaction, Latent Growth, and Monte Carlo Methods

333

The PRELIS system file, raw.psf, is displayed below, and it does contain the
three latent variables computed using the LISREL–SIMPLIS program above.

16.1.2.2 Computing Latent Interaction Variable
You create the latent interaction variable by multiplying the latent variable
scores ksi1 and ksi2. These latent variable scores are unbiased and produce
the same mean and covariance matrix as the latent variables. A PRELIS
program can be used to multiply the two independent latent variables
to create the interaction latent variable, ksi12. The PRELIS NE command
computes the latent interaction variable, which is automatically added to
the PRELIS system file, raw.psf. The CO command will treat the new latent
interaction variable as continuous rather than ordinal level of measurement. The PRELIS program is:
Create Latent Interaction Variable
SY = raw.psf
NE ksi12 = ksi1*ksi2
CO ksi12
OU RA = raw.psf

You will need to close then open the PRELIS system file, raw.psf,
before you will see the values for the interaction latent variable: ksi12. The
PRELIS raw.psf file should now contain the latent interaction variable,
ksi12, as shown below:

Note :

Y102005.indb 333

3/22/10 3:26:49 PM

334

A Beginner’s Guide to Structural Equation Modeling

You could alternatively create the latent interaction variable without running a PRELIS program. Simply, open the PRELIS system file, raw.psf, select
Transformation on the tool bar menu, then click on Compute, and underneath Add Variables, click on Add. You should see the Add Variables dialog
box. Now, simply enter the name for the latent interaction variable: ksi12.

Click OK and then the COMPUTE dialog box should appear. Now enter
the equation to create the new latent interaction variable. Follow the
instructions to drag the variable names into an equation in the Compute
dialog box. You can also obtain an equal sign (=) and a product sign (*) by
using the symbols on the calculator. Click on OK, and the latent interaction variable will instantly appear in the PRELIS system file, raw.psf.

Y102005.indb 334

3/22/10 3:26:49 PM

335

Interaction, Latent Growth, and Monte Carlo Methods

If you do not find the PRELIS System File, raw.psf, you can follow
these same steps with your own continuous variables.)

( Note :

16.1.2.3 Interaction Model Output
The PRELIS SYSTEM FILE, raw.psf, should now contain the latent interaction variable, ksi12. You can run a LISREL–SIMPLIS program to compute the coefficients (gammas) with or without an intercept term in the
structural equation. A LISREL–SIMPLIS program to compute the coefficients without an intercept term is:
Latent Interaction Variable Model - No Intercept Term
Observed Variables: V1-V9 eta ksi1 ksi2 ksi12
Raw Data from File raw.psf
Sample Size = 500
Relationships:
eta = ksi1 ksi2 ksi12
Path Diagram
End of Problem

The resultant latent variable interaction model with standardized coefficient is diagrammed in Figure 16.1b.
ksi1
0.11

0.34
0.08

ksi2

0.29

eta

0.89

–0.04

0.04

ksi12
Chi-Square = 0.00, df = 0, P-value = 1.00000, RMSEA = 0.000
Figure 16.1b
Interaction Model Output.

The structural equation from the LISREL–SIMPLIS computer output
without the intercept term is:
eta = 0.078*ksi1 + 0.16*ksi2 – 0.029*ksi12, Errorvar. = 0.21 , R² = 0.11
0.033)
(0.025)
(0.033)
(0.013)
2.36
6.36
–0.89
15.75

Y102005.indb 335

3/22/10 3:26:50 PM

336

A Beginner’s Guide to Structural Equation Modeling

16.1.2.4 Model Modification
The coefficient for the interaction latent variable, ksi12, is not statistically
significant (T = - 0.89). Therefore, you would drop this latent variable from
the model and use only ksi1 and ksi2.
The modified model output with standardized estimates would appear
as diagrammed in Figure 16.1c.
ksi1

0.10

0.34

eta
ksi2

0.89

0.29

Chi-Square = 0.00, df = 0, P-value = 1.00000, RMSEA = 0.000
Figure 16.1c
Interaction Model Modified Output.

The program produces the following output, which yields the same
R-squared value, hence the latent interaction variable did not contribute
to the prediction of eta.
16.1.2.5 Structural Equations—No Latent Interaction Variable
eta = 0.076*ksi1 + 0.16*ksi2, Errorvar.= 0.21 , R² = 0.11
(0.033)
(0.025)
(0.013)
2.31
6.35
15.76
( NO T E 1 : While the PRELIS system file is open, you can use the pulldown menu to run these models. A latent interaction variable is typically
nonnormal, even when the latent variables are normally distributed. A
solution to this problem is to use the Normal Score option in the pull down
menu for the ksi1, ksi2, and ksi12 latent variables prior to analysis.
( NO T E 2 : If eta, ksi1, ksi2, and ksi12 are variables in the PRELIS data
set, another analysis method is available. Select, STATISTICS then use
Regressions to enter the variables into the model equation, that is, RG eta
on ksi1 ksi2 ksi12. Alternatively, the RG command in a PRELIS program
can be used to conduct univariate or multivariate regression, including
ANOVA, ANCOVA, MANOVA, and MANCOVA, as well as other variations of the general multivariate linear model using a list of Y and X variables [i.e., RG Y-Varlist ON X-Varlist]).

Y102005.indb 336

3/22/10 3:26:50 PM

Interaction, Latent Growth, and Monte Carlo Methods

337

16.1.3 Two-Stage Least Squares (TSLS) Approach
Recent developments in nonlinear structural equation modeling have
focused on full information methods (e.g., maximum likelihood [ML] or
asymptotically distribution free methods [ADF or WLS]) with a concern
about estimating parameters and standard errors. We recommend bootstrap estimates of the parameters and standard errors in nonlinear models, given these estimation methods, because the observed and/or latent
interaction variables don’t meet the multivariate normality assumption.
Other problems or sources of error could exist, which is why start values
are recommended to aid convergence (i.e., the initial TSLS estimates could
be replaced with user defined start values). The two-stage least-squares
(TSLS) estimates and their standard errors are obtained without iterations and therefore provide the researcher with clues to which parameters
exceed their expected values (e.g., correlations with values greater than 1.0
in a nonpositive definite matrix). TSLS estimates therefore provide helpful
information to determine whether the specified model is reasonable.
Bollen (1995, 1996) has indicated that nonlinear SEM models can be estimated using instrumental variables in two stage least squares (TSLS). A
two-stage least squares analysis using instrumental variables is easily run
in LISREL–PRELIS (Jöreskog, Sörbom, du Toit, & du Toit, 2000, pp. 172–174)
using the following RG command (see files KJTSLS1.PR2 and KJTSLS2.PR2):
Estimating Kenny-Judd Model by Bollen’s TSLS
DA NI = 5
LA
Y X1 X2 X3 X4
RA = KJUDD.RAW
CO ALL
NE X1X3 = X1*X3
NE X1X4 = X1*X4
NE X2X3 = X2*X3
NE X2X4 = X2*X4
RG Y ON X1 X3 X1X3 WITH X2 X4 X2X4 RES=U
OU RA = KJRES.RAW

The TSLS results are as follows.
Estimated Equations
Y = 0.936 + 0.340*X1 + 0.399*X3 + 0.965*X1X3 + Error, R² = 0.594
(1.011) (0.115) (0.0883) (0.164)
0.926 2.948 4.516
5.899
The latent variable score approach is also easily run using PRELIS and
SIMPLIS programs (Jöreskog, Sörbom, du Toit, & du Toit, 2000, pp. 173;
see files KJUDD.PR2, KENJUDD.SPL, and KENJUDD.PR2). The following

Y102005.indb 337

3/22/10 3:26:51 PM

338

A Beginner’s Guide to Structural Equation Modeling

PRELIS program KJUDD.PR2, creates the PRELIS SYSTEM FILE, KJUDD.
PSF, the SIMPLIS program KENJUDD.SPL computes the latent variable
scores, and the PRELIS program KENJUDD.PR2 computes the parameter
estimates in the SEM interaction model. The PRELIS program for computing the PRELIS SYSTEM FILE is:
Computing PSF file from KJUDD.RAW
DA NI = 5
LA; Y X1 X2 X3 X4
RA = KJUDD.RAW
CO ALL
OU MA = CM RA=KJUDD.PSF

The SIMPLIS program for computing the latent variable scores is:
Estimating the Measurement Model in the Kenny–Judd Model
and Latent Variable Scores
System File from File KJUDD.DSF
Latent Variables Ksi1 Ksi2
Relationships
X1 = 1*Ksi1
X2 = Ksi1
X3 = 1*Ksi2
X4 = Ksi2
PSFfile KJUDD.PSF
Path Diagram
End of Problem

The PRELIS program for computing the parameter estimates in the SEM
interaction model is:
Estimating Kenny–Judd Model from Latent Variable Scores
SY = KJUDD.PSF
CO ALL
NE Ksi1Ksi2 = Ksi1*Ksi2
RG Y ON Ksi1 Ksi2 Ksi1Ksi2
OU

Estimated Equations
Y = 1.082 + 0.232*Ksi1 + 0.290*Ksi2 + 0.431*Ksi1Ksi2 + Error, R² = 0.381
(0.0207) (0.0297) (0.0218)
(0.0261)
Error Variance = 0.393
52.196 7.814
13.281
16.540
Interaction models comprise many different types of models. The use of
continuous variables, categorical variables, nonlinear effects, and latent
variables has intrigued scholars over the years. The current approaches
that appear easy to model are the multigroup categorical approach and

Y102005.indb 338

3/22/10 3:26:51 PM

Interaction, Latent Growth, and Monte Carlo Methods

339

the latent variables score approach, because they are not affected by many
of the problems discussed next.
The testing of interaction effects can present problems in structural equation modeling. First, you may have the problem of model specification. Linear
models simplify the task of determining relations to investigate and distributional assumptions to consider, but this may not be the case in latent variable
interaction models. Second, discarding the linearity assumption opens up
the possibility of several product indicant variable and latent variable interaction combinations, but this also serves to magnify the critical role of theory in
focusing the research effort. Third, a researcher who seeks to model categorical interaction effects must also collect data that spans the range of values in
which interaction effects are likely to be evident in the raw data, and must
collect a sample size large enough to permit subsamples. Fourth, we have
noted that the statistical fit index and parameter standard errors are based on
linearity and normality assumptions, and we may not have robust results to
recognize the presence of an interaction effect unless it is substantial.
The continuous variable approach does have its good points. It is possible to check for normality of variables, and to standardize them (Normal
Score option), and the approach does not require creating subsamples
or forming groups where observations could be misclassified, nor does
it require the researcher to categorize a variable and thereby lose information. Moreover, the continuous variable approach is parsimonious.
Basically, all but one of the additional parameters involved in the interaction model are exact functions of the main-effects parameters, so the only
new parameters to be estimated are the structure coefficient for the latent
interaction independent variable and the prediction equation error.
The continuous variable approach also has several drawbacks. First,
only a few software programs can perform the necessary nonlinear
constraints, and the programming for testing interaction effects in the
traditional sense is not easy. Second, if you include too many indicator
variables of your latent independent variables, this approach can become
very cumbersome. For example, if one latent independent variable, Factor
1, has n1 measures and the other latent independent variable, Factor 2,
has n2 measures, then the interaction term, Factor 1 x Factor 2, could have
n1 x n2 measures. If each independent latent variable has five indicator
variables, then the multiplicative latent independent variable interaction would involve 25 indicators. Including the five measures for each of
the two main-effect latent independent variables and two indicators of a
latent dependent variable, the model would have 37 indicator variables
before any other latent-variable relationships were considered. Third, the
functional form of the interaction needs to be specified. The simple multiplicative interaction presented here hardly covers other types of interactions, and for these other types of interactions there is little prior research
or available examples to guide the researcher.

Y102005.indb 339

3/22/10 3:26:51 PM

340

A Beginner’s Guide to Structural Equation Modeling

A fourth problem to consider is multicollinearity. It is very likely that
the interaction factor will be highly correlated with the observed variables
used to construct it. This multicollinearity in the measurement model
causes the interaction latent independent variable to be more highly correlated with the observed variables of other main effect latent independent
variables than each set of observed variables are with their own respective main effect latent independent variables. For multiplicative interactions between normally distributed variables, multicollinearity could be
eliminated by centering the observed variables (using scores expressed
as deviations from their means) before computing the product variable.
However, centering the variables alters the form of the interaction relationship. Researchers who want to model other types of interactions may find
no easy answer to the problem of multicollinearity (Smith & Sasaki, 1979).
A fifth concern relates to distributional problems, which are more
serious than those associated with linear modeling techniques using
observed variables only. If the observed variables are nonnormal, then
the variance of the product variable can be very different from the values implied by the basic measurement model, and the interaction effect
will perform poorly. Of course, permissible transformations may result
in a suitable, normal distribution for the observed variables. The resultant nonnormality, however, in the observed variables violates the distributional assumptions associated with the estimation methods used, for
example, maximum-likelihood. Furthermore, estimation methods that do
not make distributional assumptions may not work for interaction models. Basically, the asymptotic weight matrix associated with the covariance
matrix for an interaction model may be nonpositive definite because of
dependencies between moments of different observed variables that are
implied by the interaction model. In any case, we would recommend that
you bootstrap the parameter estimates and standard errors to achieve a
more reasonable estimate of these values.
When using the latent variable score approach you should consider
bootstrapping the standard errors because the estimation method used
may give inaccurate estimates of standard errors given violation of
the distributional assumption for the interaction model. Basically, the
asymptotic weight matrix associated with the covariance matrix for an
interaction model may be nonpositive definite because of dependencies
between moments of different observed variables that are implied by the
interaction model. In any case, we would recommend that you bootstrap
the parameter estimates and standard errors to achieve a more reasonable estimate of these values (Bollen & Stine, 1993; Mooney & Duval,
1993; Lunneborg, 1987; Stine, 1990; Jöreskog & Sörbom, 1993a; and YangWallentin & Jöreskog, 2001).
In our examples, we have assumed that the relationships in our models have been linear (i.e., the relationships among all variables, observed

Y102005.indb 340

3/22/10 3:26:51 PM

Interaction, Latent Growth, and Monte Carlo Methods

341

and latent, could be represented by linear equations). Although the use of
nonlinear and interaction effects is popular in regression models (Aiken &
West, 1991), the inclusion of interaction hypotheses in path models have been
minimal (Newman, Marchant, & Ridenour, 1993), and few examples of nonlinear factor models have been provided (McDonald, 1967; Etezadi‑Amoli
& McDonald, 1983). SEM models with interaction effects are now possible
and better understood due to several scholars including Kenny and Judd
(1984), Hayduk (1987), Wong and Long (1987), Bollen (1989), Higgins and
Judd (1990), Cole, Maxwell, Arvey, and Salas (1993), Mackenzie and Spreng
(1992), Ping (1993, 1994, 1995), Jöreskog and Yang (1996), Schumacker and
Marcoulides (1998), Algina and Moulder (2001), du Toit and du Toit (2001),
Moulder and Algina (2002), and Schumacker (2002), to name only a few.
Jöreskog and Yang (1996) do provide additional insights into modeling interaction effects, given the problems and concerns discussed here.
Jöreskog (2000) discussed many issues related to interaction modeling and
included latent variable scores in LISREL that are easy to compute and
include in interaction modeling. Schumacker (2002) compared the latent
variable score approach to the continuous variable approach using LISREL
matrix command language and found the parameter estimates to be similar
with standard errors reasonably close. Our recommendation would be to
use the latent variable score approach and bootstrap the standard errors. If
unfamiliar with the bootstrap approach, then use the Normal Score option
with interaction variables to avoid nonnormal issues when testing interaction effects.
Structural equation models that include interaction effects are not prevalent in the research literature, in part, because of all the concerns mentioned here. The categorical variable approach using multiple samples and
constraints has been used most often. The latent variable score approach
using normal scores is a useful way to model interaction with latent variables. Hopefully, more SEM research will consider interaction hypotheses
given the use of latent variable scores and the use of Normal Score data
conversion for main effect and interaction variables in LISREL–PRELIS.

16.2 Latent Growth Curve Models
Repeated measures analysis of variance has been widely used with
observed variables to statistically test for changes over time. SEM advances
the longitudinal analysis of data to include latent variable growth over
time while modeling both individual and group changes using slopes and
intercepts (McArdle & Epstein, 1987; Stoolmiller, 1995; Byrne & Crombie,
2003). Latent growth curve analysis conceptually involves two different
analyses. The first analysis is the repeated measures of each individual

Y102005.indb 341

3/22/10 3:26:51 PM

342

A Beginner’s Guide to Structural Equation Modeling

across time that is hypothesized to be linear or nonlinear. The second
analysis involves using the individual’s parameters (slope and intercept
values) to determine the difference in growth from a baseline. The latent
growth curve model (LGM) represents differences over time that takes
into account means (intercepts) and rate of change (slopes), at the individual or group level.
LGM permits an analysis of individual parameter differences, which
is critical to any analysis of change. It describes not only an individual’s
growth over time (linear or nonlinear), but also detects differences in
individual parameters over time. LGM using structural equation modeling can test the type of individual growth curve, use time varying covariates, establish the type of group curve, and include interaction effects
in latent growth curves (Li, Duncan, T.E., Duncan, S.C., Acock, YangWallentin, & Hops, 2001). The LGM approach, however, requires large
samples, multivariate normal data, equal time intervals for all subjects,
and change that occurs as a result of the time continuum (Duncan. &
Duncan, 1995).
The latent growth curve model illustrates the use of slope and intercept
as latent variables to model differences over time. The data set contains
168 adolescent responses over a 5-year period (age 11 to age 15) regarding the tolerance toward deviant behaviors, with higher scores indicating
more tolerance of such behavior. The data was transformed (i.e., log X) to
create equal interval linear measures from ordinal data. The latent growth
curve model is diagrammed in Figure 16.2a.

0
Slope

Age11

E11

Age12

E12

Age13

E13

Age14

E14

Age15

E15

1
2
3
4

1 1

1
1

Intercept

Figure 16.2a
Latent Growth Curve Model (Linear).

Y102005.indb 342

3/22/10 3:26:52 PM

Interaction, Latent Growth, and Monte Carlo Methods

343

The slope parameters are coded 0, 1, 2, 3, and 4 to establish a linear
trend with zero used as a common starting point. Other polynomial coefficients could be used for quadratic or cubic trend. The intercept parameters are coded 1, 1, 1, 1, and 1 to indicate means for the different age
groups. A LISREL–SIMPLIS program was created that shows how these
parameters are stipulated for the two latent variables, slope and intercept.
It also includes a command to correlate slope and intercept (curved arrow
in diagram) and a special term, CONST, to designate means. The LISREL–
SIMPLIS latent growth curve model program is:
16.2.1 Latent Growth Curve Program
Latent Growth Model
Observed Variables: age11 age12 age13 age14 age15
Sample size 168
Correlation matrix
1.000
.161 1.000
.408 .348 1.000
.373 .269 .411 1.000
.254 .143 .276 .705 1.000
Means .201 .226 .326 .417 .446
Standard deviations .178 .199 .269 .293 .296
Latent Variables: slope intercept
Relationships:
age11 = CONST + 0 * slope + 1 * intercept
age12 = CONST + 1 * slope + 1 * intercept
age13 = CONST + 2 * slope + 1 * intercept
age14 = CONST + 3 * slope + 1 * intercept
age15 = CONST + 4 * slope + 1 * intercept
Let slope and intercept correlate
Path Diagram
End of Problem

The initial LISREL–SIMPLIS model results indicated a poor model fit
(chi-square = 49.74, df = 7, p = 0.00). The correlation between the intercept
values (group means) and the slope (linear growth) was zero indicating
that level of tolerance at age 11 did not predict growth in tolerance across
the other age groups. However, the group means indicated otherwise, so
model modification was conducted; the means for each age are:
age11
age12
age13
age14
age15
-------- -------- -------- -------- -------0.20
0.23
0.33
0.42
0.45

Modification indices were indicated that recommended correlating the
error covariance between age 11 and age 12, as well as between age 14

Y102005.indb 343

3/22/10 3:26:52 PM

344

A Beginner’s Guide to Structural Equation Modeling

and age 15. These are apparently the two transition periods in the latent
growth curve model were more measurement disturbance was present.
16.2.2 Model Modification
The LISREL–SIMPLIS program was rerun with the following added
commands:
Let error covariance between age11 and age12 correlate
Let error covariance between age14 and age15 correlate

After modification, the latent growth curve model had a more acceptable
model fit (chi-square = 11.35, df = 5, and p = .045).
The final latent growth curve model output with standardized coefficients is diagrammed in Figure 16.2b.
Age11
0.00
Age12

Slope

0.23
0.35

Age13

0.49
–0.53

0.52
Age14

0.51
0.57

Age15

0.54

Intercept

Figure 16.2b
Latent Growth Model Output.

The individual slopes increased over time:

Y102005.indb 344

Group

Slope

Age 11
Age 12
Age 13
Age 14
Age 15

.00
.23
.35
.49
.62

3/22/10 3:26:52 PM

Interaction, Latent Growth, and Monte Carlo Methods

345

The intercepts decreased over time:
Group

Intercept

Age 11
Age 12
Age 13
Age 14
Age 15

.92
.80
.61
.57
.54

The negative correlation between the slope and intercept correctly indicates the increase in slope values over time with a corresponding decrease
in intercept values over time (r = –.53).
( Note : The LISREL–SIMPLIS computer output does not list the slope and
intercept values, but does display them in the model diagram. They were
copied and listed above for convenience.)

A test of linear rate of growth in the latent growth curve model seemed
appropriate because the means increased from .20 at age 11 to .45 at age
15. The latent growth curve model is appropriately called a Latent Growth
Curve Structured Means Model because group means as well as covariance were specified. There were individual differences in the slopes over
time. The negative correlation between the intercept values (group means),
and the slope values (linear growth) indicated that as age increased the
level of tolerance decreased.
This LGM model indicated a linear rate of growth in adolescent tolerance for deviant behavior using the age 11 as the baseline for assessing
linear change over time. You should graph these mean values across the
age levels to graphically display the trend. You should also interpret the
correlation between the intercept and slope because a positive value would
indicate that high initial status at age 11 has a greater rate of change, while
a negative correlation would indicate that high initial status at age 11 has
a lower rate of change. If the average slope value is zero, then no linear
change has occurred. Finally, you can assess how measurement errors
across adjacent years are correlated (e.g., lagged correlation in ARIMA
models). This ability to model measurement error is a unique advantage
of LGM over traditional ANOVA repeated measure designs.

16.3 Monte Carlo Methods
Researchers typically collect a random sample of data and determine if the
sample data fit a theoretical model. Model validation (chapter 12) is then
conducted to examine stability of parameter estimates and standard errors.
Generalizations are then usually made to the population parameters.

Y102005.indb 345

3/22/10 3:26:53 PM

346

A Beginner’s Guide to Structural Equation Modeling

We obviously seldom know the population model, data, or parameters,
so if we wish to investigate how statistics are affected under violations of
assumptions, etc., we need to specify the population model, generate population data (covariance matrix), and now compute model parameters to
examine how parameter estimates, standard errors, and fit indices change,
when the model is misspecified. Monte Carlo results are made easier by
writing parameter estimates, standard error estimates, and measures of
fit to a file by using PV (PV = ; stores parameter estimates), SV
(SV = ; stores standard errors); and GF (GF = , stores
goodness-of-fit indices) keywords on the LISREL OUTPUT command in
the LISREL–SIMPLIS program. The RP command permits replications,
which is also useful in Monte Carlo studies to examine how these values
change.
The Monte Carlo approach could involve simulating population data,
generating variables from a specified population covariance matrix, or
generating data from a specified model. Monte Carlo methods involve
using a pseudo-random number generator or specifying known population values to produce raw data for a population covariance matrix. (Note:
Bang, Schumacker, and Schlieve [1998] found that pseudo-random number generators do not perform the same way with many yielding nonrandom [nonnormal] distributions with sample sizes of less than 10,000).
Our interest in Monte Carlo methods is to determine the robustness of
our sample statistics, which we can only know when our population
model and/or parameters are known. The PRELIS approach to simulation of population data (covariance matrix) is described next for the path
model in Figure 16.3.
16.3.1 PRELIS Simulation of Population Data
PRELIS is considered a preprocessor for LISREL and as such screens data,
creates different types of matrices, and has other useful features for data
creation and data manipulation. PRELIS can easily produce several different types of data distributions—for example, normal and nonnormal.
We will create multivariate normally distributed population data. Simply
click on File, New, and then select PRELIS Data.

Y102005.indb 346

3/22/10 3:26:53 PM

347

Interaction, Latent Growth, and Monte Carlo Methods

V2
V1

V3
Figure 16.3
Path Model (Monte Carlo).

An empty PRELIS Data window should appear as indicated below.

We now need to insert the number of variables and the number of cases
that you want to create. We first create the number of variables, which are
four in Figure 16.3 (V1–V4). We select Data from the tool bar menu, then
Define Variables. A Define Variables dialog box appears next and then we
click on Insert.

Y102005.indb 347

3/22/10 3:26:54 PM

348

A Beginner’s Guide to Structural Equation Modeling

An Add Variables dialog box now appears and we enter the names of our
variables, that is, V1-V4., and click OK. These variables will now appear in
the Define Variables dialog box. Click OK in the Define Variables dialog box
and they will now appear in the PRELIS Data window.

Next, we select Data from the Tool Bar menu, then Insert Cases. We enter
10000 and click OK.

Y102005.indb 348

3/22/10 3:26:54 PM

Interaction, Latent Growth, and Monte Carlo Methods

349

The PRELIS Data window now appears with four variables and 10,000
cases, but with zeroes in the cells.

We are now ready to replace the zeroes with numerical values by selecting Transformation from the tool bar menu, then Compute; however, we
are first prompted to save our work as a PRELIS SYSTEM FILE (population.
psf ). (Note: Choose a directory to save the file in that will also contain your
LISREL–SIMPLIS program.)

Y102005.indb 349

3/22/10 3:26:55 PM

350

A Beginner’s Guide to Structural Equation Modeling

We can now carefully follow the directions and use the mouse to
drag and drop variables and click on n(0,1) to enter NRAND into the
equations. The equations were arbitrarily chosen to have a mean and
some correlation with other variables. (Note: Navigating this window will involve a learning curve; for example, click on Next line
to add the next variable via drag and drop. You also need to use the
mouse to enter numbers and mathematical symbols). When finished,
click OK.

After a few minutes, you will see the computed data values in the
PRELIS system file (population.psf ). Click on the save file icon to save the
data file. You can now use many of the PRELIS tool bar menu features to
calculate statistics or produces graphs of the variables.

Y102005.indb 350

3/22/10 3:26:55 PM

Interaction, Latent Growth, and Monte Carlo Methods

351

Click on Statistics in the tool bar menu, and select Output Options to
save the raw data from the PRELIS system file (population.psf ) into a covariance matrix (population.cov). (Note: A PRELIS program will appear in a
dialog box to show that a program was written to output the covariance
matrix. It will also indicate that variables are treated as continuous [CO],
provide a frequency distribution for each variable, skewness and kurtosis,
and the means and standard deviations of the V1 to V4 variables you created as population parameters).

Y102005.indb 351

3/22/10 3:26:56 PM

352

A Beginner’s Guide to Structural Equation Modeling

The covariance matrix (population.cov) can now be input into a LISREL–
SIMPLIS program (population.spl) for the model in Figure 16.3. (Note: We
wanted to treat this covariance matrix as a population matrix to obtain
population parameters, so we generated 10,000 cases.) The LISREL–
SIMPLIS program (population.spl) would be:
PRELIS Data as Population Data for Covariance Matrix
Observed variables V1 V2 V3 V4
Sample size 10000
Covariance matrix from file population.cov
Equations:
V1 = V2 V3
V4 = V1 V2 V3
Path Diagram
LISREL OUTPUT SS SC PV=parameter SV=error GF=fit
End of problem

We can now use the data set or the covariance matrix in a Monte Carlo
study to investigate what parameter estimates and standard errors might
be, given a random sample from the 10,000 cases that constitute the population. We can also save the parameter estimates (PV), standard errors
(SV), and model goodness-of-fit indices (GF) to separate files using the
LISREL OUTPUT command (Note: SS = standardized solution; SC = completely standardized solution).
Adding the LISREL OUPUT command RP = 10 would repeat the
analysis 10 times in a Monte Carlo study).

(Note :

16.3.2 Population Data from Specified Covariance Matrix
There are many different software packages that can be used to generate data
given specification of a population covariance matrix for use in Monte Carlo
studies. We chose SPSS, SAS, and LISREL matrix syntax to illustrate how to
generate population data from specification of a covariance matrix.
16.3.2.1 SPSS Approach
The SPSS MATRIX routine using the Cholesky decomposition can be used to
generate raw data and output an SPSS save file. The SPSS save file can then
be imported into a LISREL program. The following SPSS MATRIX program
only requires the population covariance or correlation matrix (r), sample
size (n), and output file name, Save . (Note: Save corr/outfile = *. ;
it will output data into an SPSS Untitled dialog box which you can then save
as an SPSS save file.) The SPSS MATRIX program requires a symmetrical
matrix as input. To execute the SPSS MATRIX program, open SPSS, select
File, New, and then Syntax. Enter the SPSS Matrix program into the syntax

Y102005.indb 352

3/22/10 3:26:56 PM

Interaction, Latent Growth, and Monte Carlo Methods

353

window, save it, and then click on the run command on the tool bar menu to
execute the program and save the data into an SPSS save file, sample.sav.
MATRIX.
compute popr =
{1, .4, .3, .2;
.4, 1, .6, .7;
.3, .6, 1, .8;
.2, .7, .8, 1}.
Print popr.
compute pi = 3.14159.
compute rown = nrow(popr).
compute n = 10000.
compute corr = sqrt(-2*ln(uniform(n,rown)))&*cos((2*pi)*
uniform(n,rown)).
compute corr=corr*chol(popr).
save corr /outfile = pop.sav.
END MATRIX.

We had the SPSS Matrix program print the popr matrix to verify it was
read correctly. The SPSS output should look like the following:
Run MATRIX procedure:
POPR
1.000000000 .400000000 .300000000 .200000000
.400000000 1.000000000 .600000000 .700000000
.300000000 .600000000 1.000000000 .800000000
.200000000 .700000000 .800000000 1.000000000
------ END MATRIX -----

We can now open the SPSS save file, pop.sav, and compute the bivariate
correlation between the variables, COL1 – COL4, which can be renamed,
if desired.

Y102005.indb 353

3/22/10 3:26:57 PM

354

A Beginner’s Guide to Structural Equation Modeling

The correlation values obtained from SPSS bivariate correlation routine
will approximate the ones specified in the popr matrix. (Note: The correlations should be within .01 of the population correlation/covariance values. Also, every time you run the SPSS MATRIX program you will get
slightly different values for the correlations, unless using a random seed
number.) The SPSS Correlation Output is in Table 16.1.
Table 16.1
SPSS Correlation Output
Correlations
COLUMN 1
COL1

Pearson Correlation
Sig. (two-tailed)
N

1.000
10000.000

COLUMN 2

.304
.000
10000

.201a
.000
10000

1.000

.593a
.000
10000

.694a
.000
10000

1.000

.800a
.000
10000

Pearson Correlation
Sig. (two-tailed)
N

.404a
.000
10000

COL3

Pearson Correlation
Sig. (two-tailed)
N

.304
.000
10000

.593a
.000
10000

Pearson Correlation
Sig. (two-tailed)
N

.201a
.000
10000

.694a
.000
10000

COL4

COLUMN 4

.404
.000
10000

COL2

COLUMN 3
a

10000.000

10000.000
.800a
.000
10000

1.000
10000.000

Correlation is significant at the 0.01 level (two-tailed).

16.3.2.2 SAS Approach
A SAS program can also be written to produce data from a population
covariance matrix using a normal distribution function. The SAS program
is written as:
proc iml;
cov = {1 .4 .3 .2,
.4 1 .6 .7,
.3 .6 1 .8,
.2 .7 .8 1};
print cov;
v = nrow(cov);
n = 10000;
seed = 12345;
l = t(root(cov));

Y102005.indb 354

/* Generate multivariate normal data in SAS/IML */

/* population correlation matrix */
/* calculate number of variables */
/* input number of cases */
/* random seed number */
/* calculate cholesky root of cov matrix */

3/22/10 3:26:57 PM

355

Interaction, Latent Growth, and Monte Carlo Methods

z = normal(j(v,n, seed));
x = l*z;
tx = t(x);
create cor from tx;
append from tx;
quit;
Proc corr data = cor;
var col1 col2 col3 col4;
run;

/* generate nvars*samplesize normal distribution */
/* premultiply by cholesky root */
/* transpose of X */
/* write out sample data to sas dataset */

/* sample covariance matrix */

The SAS population matrix and the sample matrix from Proc corr should
be similar, as desired. Changing the seed number, however, will produce
slightly different results each time you run the SAS program. Our SAS 9.1
computer output looked like:
COV
1 0.4 0.3 0.2
0.4
1 0.6 0.7
0.3 0.6
1 0.8
0.2 0.7 0.8
1
The CORR Procedure
4 Variables: COL1 COL2 COL3 COL4
Simple Statistics
Variable
COL1
COL2
COL3
COL4

N
Mean
Std Dev
Sum
Minimum Maximum
10000 0.00591 1.00676 59.12856 −4.06923 4.13280
10000 −0.00628 1.00321 −62.81136 −4.47883 3.55955
10000 0.01407 1.00337 140.74048 −3.50194 3.81102
10000 0.00662 0.99853 66.17666 −3.45835 3.63828

Pearson Correlation Coefficients, N = 10000
Prob > |r| under H0: Rho=0
COL1
COL2
COL3
COL4

COL1
1.00000
0.41708
0.32340
0.22719

COL2
0.41708
1.00000
0.60942
0.70242

COL3
0.32340
0.60942
1.00000
0.80413

COL4
0.22719
0.70242
0.80413
1.00000

16.3.2.3 LISREL Approach
It is also possible to generate multivariate normal variables with a desired
population covariance matrix using either the Cholesky decomposition or

Y102005.indb 355

3/22/10 3:26:57 PM

356

A Beginner’s Guide to Structural Equation Modeling

factor pattern matrix approach in LISREL. We will first input four variables
and use the Cholesky decomposed matrix of coefficients to compute four
new variables with the desired covariance structure. Secondly, we will
use a pattern matrix approach to generate the same Cholesky decomposed matrix of coefficients that one would use to compute the same new
multivariate normal variables.
16.3.2.3.1 Cholesky Decomposition Approach
Cholesky decomposition of our symmetric population covariance matrix,
S, yields a Lambda Y matrix. The coefficients in the Lambda Y matrix are
then used to compute the new variables. You can save either a covariance matrix (RS option) or raw data (RA option); we saved a covariance
matrix (POP.CM). You will need to run a series of programs to accomplish the generation of the multivariate normally distributed data for
your variables.
Program 1 is a LISREL matrix program which inserts a specified population covariance matrix (CM) with the number of variables, Y1–Y4 (LE),
indicated for a model (MO) that has the Lambda Y values to be freely estimated (FR). The model must be saturated (c2 = 0) and the residual errors
set to zero (TE = ZE). The resulting Lambda Y matrix provides the coefficients to be used to compute the new multivariate normal variables, V1–V4
(LA). (Note: You must specify, all Y variables and associated matrices in
the MO command line; or correspondingly, all X variables and associated
matrices; but not a mix of X and Y variables and associated matrices or the
program will not work.)
Program 1
! Cholesky decomposition matrix approach
DA NI = 4 NO = 10000
LA
V1 V2 V3 V4
CM
1.000
0.41708 1.000
0.32340 0.60942 1.000
0.22719 0.70242 0.80413 1.000
MO NY = 4 NE = 4 LY = FU,FI BE = FU,FI PS = SY,FI TE =ZE
LE
Y1 Y2 Y3 Y4
VA 1.0 PS (1, 1) PS (2, 2) PS (3, 3) PS (4, 4)
FR LY (1, 1) LY (2, 2) LY (3, 3) LY (4, 4)
FR LY (2, 1) LY (3, 1) LY (4, 1)
FR LY (3, 2) LY (4, 2)
FR LY (4, 3)
OU ND = 5 RS

Y102005.indb 356

3/22/10 3:26:57 PM

357

Interaction, Latent Growth, and Monte Carlo Methods

LAMBDA-Y

V1
V2
V3
V4

Y1
-------1.00000
0.41708
0.32340
0.22719

Y2
-------- 0.90887
0.52212
0.66859

Y3
-------- - 0.78918
0.48351

Y4
-------- - - 0.51729

Program 2 uses the Lambda Y values in a PRELIS program to compute
the new variables, V1-V4. The Y1–Y4 variables are first generated from normally distributed random data (NRAND function) using a seed value (IX =
12345). Next, new variables are created for V1-V4 using the coefficients from
the Lambda Y matrix and saved in a covariance matrix (CM = POP.CM).
The Y1-Y4 variables are deleted (SD). The RA = option would
save raw data for the variables instead of a matrix if so desired.
Program 2
! Compute new multivariate normal variables from Lambda Y
matrix
DA NO = 10000
NE Y1 = NRAND
NE Y2 = NRAND
NE Y3 = NRAND
NE Y4 = NRAND
NE V1 = 1 * Y1
NE V2 = .41708 * Y1 + .90887 * Y2
NE V3 = .32340 * Y1 + .52212 * Y2 + .78918 * Y3
NE V4 = .22719 * Y1 + .66859 * Y2 + .48351 * Y3 + .51729* Y4
CO ALL
SD Y1-Y4
OU CM = POP.CM ND = 5 XM IX = 12345

Finally, Program 3 would run a LISREL–SIMPLIS program with the generated population covariance matrix to produce the specified model in
Figure 16.3.
Program 3
Path model Figure 16.3 with Cholesky decomposed matrix
variables
Observed variables V1 V2 V3 V4
Sample size 10000
!Covariance Matrix from file POP.CM
Covariance Matrix
0.99641
0.42637 1.0185
0.32652 0.62854 1.0379
0.23881 0.72385 0.83883 1.0322

Y102005.indb 357

3/22/10 3:26:57 PM

358

A Beginner’s Guide to Structural Equation Modeling

Equation:
V1 = V2 V3
V4 = V1 V2 V3
Number of Decimals = 5
Path Diagram
End of Problem

We used sample size of 10,000 and 5 decimal places to avoid
rounding error and non-convergence problems.)

(Note :

16.3.2.3.2 Pattern Matrix Approach
The pattern matrix approach is possible by inputting the pattern matrix
(PA) and corresponding lambda X matrix (MA) with the specified covariance matrix (CM). The results would be the same as before. The Lambda
X coefficients would be the same as before and used in Program 2 above
to compute multivariate normal variables. The LISREL program would be
written as:
! Pattern Matrix approach
DA NI = 4 NO = 10000
LA
V1 V2 V3 V4
CM
1.000
0.41708 1.000
0.32340 0.60942 1.000
0.22719 0.70242 0.80413 1.000
MO NX = 4 NK = 4 PH = ID TD =ZE
PA LX
1 0 0 0
1 1 0 0
1 1 1 0
1 1 1 1
MA LX
1 0 0 0
1 1 0 0
1 1 1 0
1 1 1 1
OU ND = 5 RS
LAMBDA-X

V1
V2
V3
V4

Y102005.indb 358

KSI 1
-------1.00000
0.41708
0.32340
0.22719

KSI 2
-------- 0.90887
0.52212
0.66859

KSI 3
-------- - 0.78918
0.48351

KSI 4
-------- - - 0.51729

3/22/10 3:26:57 PM

Interaction, Latent Growth, and Monte Carlo Methods

359

( Note : It is also straightforward to compute the Cholesky decomposed
matrix using SPSS to check your programming. The SPSS MATRIX procedure with the original population covariance matrix used (S) and the
resulting Cholesky decomposed matrix [SCHOL] is output as follows.)

MATRIX.
Compute S = {1.00000, .41708, .32340, .22719;
.41708, 1.00000, .60942, .70242;
.32340, .60942, 1.00000, .80413;
.22719, .70242, .80413, 1.00000}.
Print S.
Compute SCHOL = T(CHOL(S)).
Print SCHOL.
END MATRIX.

Run MATRIX procedure:
S
1.000000000 .417080000 .323400000 .227190000
.417080000 1.000000000 .609420000 .702420000
.323400000 .609420000 1.000000000 .804130000
.227190000 .702420000 .804130000 1.000000000
SCHOL
1.000000000 .000000000 .000000000 .000000000
.417080000 .908869778 .000000000 .000000000
.323400000 .522116963 .789180789 .000000000
.227190000 .668592585 .483505465 .517292107

16.3.3 Covariance Matrix from Specified Model
A more appropriate way to generate a population covariance matrix is
from a specified population model. This permits a better way to examine how model misspecification affects overall model fit as well as predefined population parameter values. Unfortunately, the population
model specification and subsequent generation of population model
parameters is not directly possible using LISREL or PRELIS programs.
The reason is that not all matrices, especially covariance and certain error
terms, can be specified in the programs. The solution is to (1) specify a
population model, (2) define what matrices are indicated in the population model, (3) pick values for the population parameters in the matrices,
and then (4) use matrix operations to compute the population covariance
matrix. In a final step (5), you can verify that the population model with
the population parameters was correctly specified by using the population covariance matrix in a LISREL–SIMPLIS program. We will now take
you through these steps to illustrate a better way to conduct Monte Carlo

Y102005.indb 359

3/22/10 3:26:57 PM

360

A Beginner’s Guide to Structural Equation Modeling

X6
Figure 16.4
Population Model (Monte Carlo).

studies (rather than simulation of data or generation of data from a specified correlation/covariance matrix).
Step 1: We begin by specifying a population model in Figure 16.4. The
population model is a confirmatory factor model with two latent independent factors, x1 and x2. Each of the latent independent variables is
measured by three indicator (observed) variables. The indicator variables X1 to X3 define the first latent independent variable and indicator
variables X4 to X6 define the second latent independent variable. Each of
the indicator variables have measurement error, designated as: d 1 to d 6.
We also have lambda X values or factor loadings for each of the paths
from the latent independent variables to the indicator variables. Finally,
we have to specify the covariance between the two latent independent
variables.
Step 2: We define what matrices are indicated in this population model.
We described a lambda X matrix (factor loadings), a theta–delta matrix
(measurement error of indicator variables), and a phi matrix (correlation
between the two factors).
Step 3: We specify what we want our population parameters to be in
these matrices. We chose to set factor loadings for X1 to X3 at .6 and factor
loadings for X4 to X6 at .7. We then calculated our measurement errors as
1 − (.6)2 for X1 to X3 and 1 – (.7)2 for X4 to X6; .64 and .51, respectively, in

Y102005.indb 360

3/22/10 3:26:58 PM

361

Interaction, Latent Growth, and Monte Carlo Methods

the theta–delta matrix. (Note: Failure to have factor loadings and measurement error synchronized—that is, X = T + E—will lead to a nonpositive
definite matrix and error warning.) Finally, we set the factor correlation at
.70. The matrices with our selected population parameters for the confirmatory factor model would be as follows:
.6

.6
.6
Λx = 
0
0

 0

.64
0


0
0
0
0
 Θδ = 
.7 
0

0
.7


.7 
 0

0
.64
0
0
0
0

0
0
.64
0
0
0

0
0
0
.51
0
0

0
0
0
0
.51
0

0 

0 
1.0
0 
Φ = 
0 
 .7

0

.51

.7 

1.0 

Please be aware that these matrices in a LISREL program with a pattern
matrix or MO commands will not create a population covariance matrix
because we cannot specify the measurement errors of the indicator variables
exactly (typically created with random number generator) nor the correlation
between the factors. Also, there are two other implied matrices that would
have zero values: tau matrix of zero mean values for indicator variables (t x )
and alpha matrix for means of our latent independent variables (a), although
these are not used in our calculations of the population covariance matrix
that are implicitly set to zero. These two matrices are indicated as:
0
 
0
0
0
τ x =  α =  
0
0
0
 
 0 
Step 4: We now use matrix operations with these matrices to produce the
population covariance matrix (Σ). The covariance matrix equation would
multiply the coefficients in the lambda X matrix (LX) times the phi matrix
(phi) and post multiply times the transpose of the lambda X matrix (LXT),
plus add the measurement error of each indicator variable, which is represented as:
Σ = Λ X ΦΛ ′X + Θδ

Y102005.indb 361

3/22/10 3:26:59 PM

362

A Beginner’s Guide to Structural Equation Modeling

We used the SPSS MATRIX procedure to compute the population covariance matrix, which uses full symmetric matrices. The SPSS MATRIX
program is:
Matrix.
compute LX= {.6,.0;
.6,.0;
.6,.0;
.0,.7;
.0,.7;
.0,.7}.
print LX.
compute phi = {1,.7;
.7 ,1}.
print phi.
compute thetad={.64,0,0,0,0,0;
0,.64,0,0,0,0;
0,0,.64,0,0,0;
0,0,0,.51,0,0;
0,0,0,0,.51,0;
0,0,0,0,0,.51}.
print thetad.
compute LXT = T(LX).
print LXT.
compute sigma = LX * phi * LXT + thetad.
print sigma.
end matrix.

The resulting output with a lambda matrix of factor loadings (LX), phi
matrix with factor correlation (PHI), theta–delta matrix with measurement errors for the indicator variables (THETAD), transpose of LX matrix
(LXT) are indicated below, along with the population covariance matrix
(SIGMA):
Run MATRIX procedure:
LX
.6000000000 .0000000000
.6000000000 .0000000000
.6000000000 .0000000000
.0000000000 .7000000000
.0000000000 .7000000000
.0000000000 .7000000000
PHI
1.000000000 .700000000
.700000000 1.000000000

Y102005.indb 362

3/22/10 3:26:59 PM

Interaction, Latent Growth, and Monte Carlo Methods

THETAD
.6400000000
.0000000000
.0000000000
.0000000000
.0000000000
.0000000000

.0000000000
.6400000000
.0000000000
.0000000000
.0000000000
.0000000000

.0000000000
.0000000000
.6400000000
.0000000000
.0000000000
.0000000000

.0000000000
.0000000000
.0000000000
.5100000000
.0000000000
.0000000000

.0000000000
.0000000000
.0000000000
.0000000000
.5100000000
.0000000000

363

0000000000
.0000000000
.0000000000
.0000000000
.0000000000
.5100000000

LXT
.6000000000 .6000000000 .6000000000 .0000000000 .0000000000 .0000000000
.0000000000 .0000000000 .0000000000 .7000000000 .7000000000 .7000000000
SIGMA
1.000000000 .360000000 .360000000 .294000000 .294000000 .294000000
.360000000 1.000000000 .360000000 .294000000 .294000000 .294000000
.360000000 .360000000 1.000000000 .294000000 .294000000 .294000000
.294000000 .294000000 .294000000 1.000000000 .490000000 .490000000
.294000000 .294000000 .294000000 .490000000 1.000000000 .490000000
.294000000 .294000000 .294000000 .490000000 .490000000 1.000000000
------ END MATRIX -----

Step 5: We now include the population covariance matrix (SIGMA) in a
LISREL–SIMPLIS program to produce the population confirmatory factor
model (Figure 16.5) that should indicate the values we picked for the population parameters. We only need to include the lower triangular matrix
in the program. The LISREL–SIMPLIS program with our SIGMA (Σ) covariance matrix is:
Confirmatory Factor Model in Figure 16.5
Observed variables X1 X2 X3 X4 X5 X6
Sample size 1000
Covariance Matrix
1.00000
.360000 1.00000
.360000 .360000 1.00000
.294000 .294000 .294000 1.00000
.294000 .294000 .294000 .490000 1.000000
.294000 .294000 .294000 .490000 .490000 1.00000
Latent variables KSI1 KSI2
Relationships:
X1 - X3 = KSI1
X4 - X6 = KSI2
Number of Decimals = 5
Path Diagram
End of Problem

Figure 16.5 does indeed show the factor loadings, factor correlation,
and measurement error for the indicator variables we specified for our

Y102005.indb 363

3/22/10 3:26:59 PM

364

A Beginner’s Guide to Structural Equation Modeling

0.64

0.60
0.60
0.60

KSI1

0.70
0.51

0.51

0.70
0.70

KSI2

0.70

Chi-Square = 0.00, df = 8, P-value = 1.00000, RMSEA = 0.000
Figure 16.5
Confirmatory Factor Model (Monte Carlo).

population confirmatory factor model. Please note that the model fits perfectly (c2 = 0, p = 1). We could now generate data using this population
covariance matrix in a SAS or SPSS program and introduce various misspecified models to determine how our population model is affected—for
example, model-fit indices, parameters, and standard errors change.
In SEM, the use of a known population covariance matrix permits a
comparison with alternative models that produce differing implied covariance matrices that can be compared with the population covariance
matrix, as well as an examination of the parameter estimates from the
implied model. The variation in the covariance matrices, parameter estimates, and fit indices can be inspected given the new LISREL OUTPUT
optional commands; PV, SV, and GF.
Monte Carlo studies are typically conducted to examine model fit,
parameter estimates, and standard errors to determine how much
they fluctuate or change under certain conditions, for example, different sample sizes, missing data, and/or nonnormal distributions. More
complex programs are possible that use data generated from different types of distributions using optional commands other than normal

Y102005.indb 364

3/22/10 3:27:00 PM

Interaction, Latent Growth, and Monte Carlo Methods

365

when generating data. For example, Enders and Bandalos (2001) studied
a three-factor model to compare four methods of missing data estimation. Their experimental condition included different factor loadings,
sample size, percent missingness of data, and type of data missingness
(MCAR or MAR).
Conducting Monte Carlo studies however can be cumbersome. Paxton,
Curran, Bollen, Kirby, and Chen (2001) provided useful steps to follow when conducting a Monte Carlo study. Their basic steps are outlined below, but the reader is encouraged to read their complete journal
article.
Step 1: Develop a research question.
Step 2: Create a valid model.
Step 3: Select experimental conditions.
Step 4: Select values of population parameters.
Step 5: Select software package.
Step 6: Conduct simulations.
Step 7: File storage requirements.
Step 8: Troubleshoot and verify results.
Step 9: Summarizing results.
Mooney (1997) provides a basic introduction to Monte Carlo simulation.
Skrondal (2000) and Paxton et al. (2001) also offer advice on Monte Carlo
methods. Fan, Felsovalyi, Sivo, and Keenan (2002) have written an excellent guide for quantitative researchers who wish to conduct Monte Carlo
studies using SAS; a Web site is provided to download a zip file with SAS
Monte Carlo programs. Fan (2005) has also published a “Teachers Corner”
article on using SAS in structural equation modeling. Bandalos (2006) provides SEM examples generating data from specified models. Long (2008)
additionally provided helpful suggestions associated with managing
data, although using STATA software, the data management suggestions
are helpful. We recommend following their suggestions when conducting
a Monte Carlo study.

16.4 Summary
In this chapter we have presented many different types of interaction
models. The use of continuous variables, categorical variables, nonlinear
effects, and latent variables has provided the basis for discussing different

Y102005.indb 365

3/22/10 3:27:00 PM

366

A Beginner’s Guide to Structural Equation Modeling

interaction models. The two current variable interaction approaches that
are easy to model would be the latent variables score approach using
normal scores or two-stage least squares approach. We highly recommend either of these two options. Our discussion of latent growth curve
models introduced SEM applications for longitudinal data analysis of
latent variables. Today, more and more emphasis is being placed on longitudinal data analysis and models. It is a logical extension to expand
our thinking into the use of latent variables when applying longitudinal models that heretofore had only used observed variables. Finally,
we presented Monte Carlo methods because of the usefulness in creating population models that then permit examination of how parameter
estimates, standard errors, and fit indices are affected by missing data
sample size, nonnormality, distribution assumptions, and other factors
that affect statistical estimation. Specifically, we examined four ways to
obtain population data and/or correlation/covariance matrix: (1) simulation of population data, (2) Cholesky decomposition of a specified population matrix to obtain parameter coefficients, (3) pattern matrix of a
specified population matrix, which can also be used to obtain parameter coefficients, and finally, (4) obtaining population covariance matrix
from a specified population model. We hope these SEM methods have
enhanced further your understanding of the usefulness of structural
equation modeling.
Exercises
1. Interaction Model
An organizational psychologist was investigating whether work
tension and collegiality were predictors of job satisfaction. However,
research indicated that work tension and collegiality interact, so a SEM
Interaction Model was hypothesized and tested. The Interaction
Model is diagrammed in Figure 16.6.
Use LISREL to OPEN the PRELIS system file, jobs.psf, then proceed
to follow the necessary steps to create the latent variables (work tension, collegiality, and job satisfaction, interaction) and add them to
the PRELIS system file. Next, create and run a LISREL–SIMPLIS program to test the interaction model. What conclusions can you make
regarding the interaction of the latent variables work tension and
collegiality?
2. Latent Growth Curve Model
News and radio stations in Dallas, Texas, have apparently convinced
the public that a massive crime wave has occurred during the past
4 years, from 2002 to 2005. A criminologist gathered the crime rate
data, but needs your help to run a latent growth curve model to test

Y102005.indb 366

3/22/10 3:27:00 PM

Interaction, Latent Growth, and Monte Carlo Methods

367

V4
Work Tension

gamma1
V1
Work Tension
×
Collegiality

gamma3

Job Satisfaction

gamma2

Collegiality

V9
Figure 16.6
Job Satisfaction Exercise.

whether a linear trend in crime rates exist for the city. The data set
information is:
Observed variables: time1 time2 time3 time4
Sample Size 400
Correlation Matrix
1.000
.799 1.000
.690 .715 1.000
.605 .713 .800 1.000
Means 5.417 5.519 5.715 5.83
Standard Deviations .782 .755 .700 .780

Create a LISREL–SIMPLIS program, diagram the model with standardized coefficients and interpret your findings. Have crime rates
increased in Dallas from 2002 to 2005?

Y102005.indb 367

3/22/10 3:27:00 PM

368

A Beginner’s Guide to Structural Equation Modeling

3. Monte Carlo Method
Write a program using either SPSS or SAS for the following population matrix and generate data (N = 10,000 cases):
1.00
.50 1.00
.30 .70 1.00
.90 .50 .50 1.00

Given the generated data, compute the population correlation matrix.
Does the generated data recover the population correlation values in
the matrix?

Suggested Readings
Interaction Models
Fielding, D., & Torres, S. (2005). A simultaneous equation model of economic development and income inequality. Journal of Economic Inequality, 4, 279–301.
Ritchie, M. D., Hahn, L. W., Roodi, N., Bailey, L. R., Dupont, W. D., Parl, F. F., &
Moore, J. H. (2001). Multifactor-dimensionality reduction reveals high-order
interactions among estrogen-metabolism genes in sporadic breast cancer.
American Journal of Human Genetics, 69, 138–147.
Schumacker, R. E. (2002). Latent variable interaction modeling. Structural Equation
Modeling: A Multidisciplinary Journal, 9, 40–54.

Latent Growth-Curve Models
Duncan, S. C., & Duncan, T. E. (1994). Modeling incomplete longitudinal substance use data using latent variable growth curve methodology. Multivariate
Behavioral Research, 29(4), 313–338.
Ghisletta, P., & McArdle, J. J. (2001). Latent growth curve analyses of the development of height. Structural Equation Modeling: A Multidisciplinary Journal, 8,
531–555.
Shevlin, M., & Millar, R. (2006). Career education: An application of latent growth
curve modeling to career information-seeking behavior of school pupils.
British Journal of Educational Psychology, 76, 141–153.

Monte Carlo Methods
Stephenson, M. T., & Holbert, R. L. (2003). A Monte Carlo simulation of observable versus latent variable structural equation modeling techniques.
Communication Research, 30(3), 332–354.
Fan, X. (2005). Using SAS for Monte Carlo simulation research in SEM. Structural
Equation Modeling: A Multidisciplinary Journal, 12(2), 299–33.

Y102005.indb 368

3/22/10 3:27:01 PM

Interaction, Latent Growth, and Monte Carlo Methods

369

Muthèn, L. K., & Muthèn, B. O. (2002). Teacher’s corner: How to use a Monte Carlo
study to decide on sample size and determine power. Structural Equation
Modeling: A Multidisciplinary Journal, 9(4), 599–620.

References
Aiken, L. S., & West, S. G. (1991). Multiple regression: Testing and interpreting interactions. Newbury Park, CA: Sage.
Algina, J., & Moulder, B. C. (2001). A note on estimating the Jöreskog-Yang model
for latent variable interaction using LISREL 8.3. Structural Equation Modeling,
8(1), 40–52.
Bandalos, D. L. (2006). The use of Monte Carlo studies in structural equation modeling. In G. R. Hancock & R. O. Mueller (Eds.), Structural equation modeling: A
second course (pp. 385–426). Greenwich, CT: Information Age.
Bang, J. W., Schumacker, R. E., & Schlieve, P. L. (1998). Random-number generator
validity in simulation studies: An investigation of normality. Educational and
Psychological Measurement, 58(3), 430–450.
Bollen, K. A. (1989). Structural equations with latent variables. New York: John Wiley
& Sons.
Bollen, K. A., & Stine, R. A. (1993). Bootstrapping goodness-of-fit measures in
structural equation models. In K. A. Bollen & J. S. Long (Eds.), Testing structural equation models (pp. 66–110). Newbury Park, CA: Sage.
Bollen, K. A. (1995). Structural equation models that are nonlinear in latent variables: A least squares estimator. In P. M. Marsden (Ed.), Sociological methodology 1995. Cambridge, MA: Blackwell.
Bollen, K. A. (1996). An alternative two stage least squares (2SLS) estimator for
latent variable equations. Psychometrika, 61, 109–121.
Byrne, B. M. & Crombie, G. (2003). Modeling and testing change : an introduction
to the latent growth curve model. Understanding Statistics, 2(3), 177–203.
Cole, D. A., Maxwell, S. E., Arvey, R., & Salas, E. (1993). Multivariate group comparisons of variable systems: MANOVA and structural equation modeling.
Psychological Bulletin, 114, 174–184.
Duncan, T. E., & Duncan, S. C. (1995). Modeling the processes of development via
latent variable growth curve methodology. Structural Equation Modeling, 2(3),
187–213.
Du Toit, M., & du Toit, S. (2001). Interactive LISREL: User’s guide. Lincolnwood, IL:
Scientific Software International.
Enders, C. K., & Bandalos, D. L. (2001). The relative performance of full information maximum likelihood estimation for missing data in structural equation
models. Structural Equation Modeling, 8, 430–457.
Etezadi‑Amoli, J., & McDonald, R. P. (1983). A second generation nonlinear factor
analysis. Psychometrika, 48, 315–342.
Fan, X., Felsovalyi, A., Sivo, S., & Keenan, S. C. (2002). SAS for Monte Carlo Studies:
A Guide for Quantitative Researchers. Sage Publications: CA. [zip file of SAS
Monte Carlo programs available at: http://support.sas.com/publishing/
bbu/57323/57323.zip].

Y102005.indb 369

3/22/10 3:27:01 PM

370

A Beginner’s Guide to Structural Equation Modeling

Fan, X (2005). Using SAS for Monte Carlo simulation research in SEM. Structural
Equation Modeling: A Multidisciplinary Journal, 12(2), 299–33.
Hayduk, L. A. (1987). Structural equation modeling with LISREL. Baltimore, MD:
Johns Hopkins University Press.
Higgins, L. F., & Judd, C. M. (1990). Estimation of non-linear models in the presence of measurement error. Decision Sciences, 21, 738–751.
Jöreskog, K. G. (2000). Latent variable scores and their uses. Lincolnwood, IL: Scientific
Software International.
Jöreskog, K. G., & Sörbom, D. (1993a). Bootstrapping and Monte Carlo experimenting
with PRELIS2 and LISREL8. Chicago, IL: Scientific Software International.
Jöreskog, K. G., & Sörbom, D. (1993b). LISREL8 user’s reference guide. Chicago, IL:
Scientific Software International, Inc.
Jöreskog, K. G., & Sörbom, D. (1993c). LISREL 8: Structural equation modeling with
the SIMPLIS command language. Chicago: Scientific Software International.
Jöreskog, K. G., & Sörbom, D. (1993d). PRELIS2 user’s reference guide. Chicago, IL:
Scientific Software International.
Jöreskog, K. G., Sörbom, D., Du Toit, S., & Du Toit, M. (2000). LISREL8: New statistical features. Lincolnwood, IL: Scientific Software International.
Jöreskog, K. G., & Yang, F. (1996). Non-linear structural equation models: The
Kenny-Judd model with interaction effects. In G. A. Marcoulides, & R. E.
Schumacker (Eds.), New developments and techniques in structural equation modeling (pp. 57–88). Mahwah, NJ: Lawrence Erlbaum.
Kenny, D. A., & Judd, C. M. (1984). Estimating the non‑linear and interactive effects
of latent variables. Psychological Bulletin, 96, 201–210.
Li, F., Duncan, T. E., Duncan, S. C., Acock, A. C., Yang-Wallentin, F., & Hops, H.
(2001). Interaction models in latent growth curves. In G. A. Marcoulides, &
R. E. Schumacker (Eds.), New developments and techniques in structural equation modeling (pp. 173–201). Mahwah, NJ: Lawrence Erlbaum.
Long, J. S. (2008). The workflow of data analysis using STATA. College Station, TX:
Stata Press.
Lunneborg, C. E. (1987). Bootstrap applications for the behavioral sciences: Vol. 1.
Psychology Department, University of Washington, Seattle.
Mackenzie, S. B., & Spreng, R. A. (1992). How does motivation moderate the
impact of central and peripheral processing on brand attitudes and intentions? Journal of Consumer Research, 18, 519–529.
McArdle, J. J., & Epstein, D. (1987). Latent growth curves within developmental
structural equation models. Child Development, 58, 110–133.
McDonald, R. P. (1967). Nonlinear factor analysis. Psychometric Monograph,
No. 15.
Moulder, B. C., & Algina, J. (2002). Comparison of method for estimating and testing latent variable interactions. Structural Equation Modeling, 9(1), 1–19.
Mooney, C. Z., & Duval, R. D. (1993). Bootstrapping: A nonparametric approach to
statistical inference. Sage University Series on Quantitative Applications in the
Social Sciences, 07-097. Beverly Hills, CA: Sage.
Mooney, C. Z. (1997). Monte Carlo Simulation. Sage Series on Quantitative
Applications in the Social Sciences. Beverly Hills, CA: Sage.

Y102005.indb 370

3/22/10 3:27:01 PM

Interaction, Latent Growth, and Monte Carlo Methods

371

Newman, I., Marchant, G. J., & Ridenour, T. (1993, April). Type VI errors in path
analysis: Testing for interactions. Paper presented at the annual meeting of the
American Educational Research Association, Atlanta.
Paxton, P., Curran, P. J., Bollen, K. A., Kirby, J., & Chen, F. (2001). Monte Carlo experiments: Design and implementation. Structural Equation Modeling, 8, 287–312.
Ping, R. A., Jr. (1993). Latent variable interaction and quadratic effect estimation: A suggested approach. Technical Report. Dayton, OH: Wright State University.
Ping, R. A., Jr. (1994). Does satisfaction moderate the association between alternative attractiveness and exit intention in a marketing channel? Journal of the
Academy of Marketing Science, 22(4), 364–371.
Ping, R. A., Jr. (1995). A parsimonious estimating technique for interaction and
quadratic latent variables. Journal of Marketing Research, 32(3), 336–347.
Schumacker, R. E., & Marcoulides, G. A. (1998). Interaction and nonlinear effects in
structural equation modeling. Mahwah, NJ: Lawrence Erlbaum.
Schumacker, R. E., & Rigdon, E. (1995, April). Testing interaction effects in structural
equation modeling. Paper presented at the annual meeting of the American
Educational Research Association, San Francisco.
Schumacker, R. E. (2002). Latent variable interaction modeling. Structural Equation
Modeling, 9(1), 40–54.
Skrondal, A. (2001). Design and analysis of Monte Carlo experiments: Attacking
the conventional wisdom. Multivariate Behavioral Research, 35, 137–167.
Smith, K. W., & Sasaki, M. S. (1979). Decreasing multicollinearity: A method for
models with multiplicative functions. Sociological Methods and Research, 8,
35–56.
Stine, R. (1990). An introduction to bootstrap methods: Examples and ideas. In J.
Fox. & J. S. Long (Eds.), Modern methods of data analysis (pp. 325–373). Beverly
Hills, CA: Sage.
Stoolmiller, M. (1995). Using latent growth curves to study developmental processes. In J. M. Gottman (Ed.), The analysis of change (pp. 103–138). Mahwah,
NJ: Lawrence Erlbaum.
Wong, S. K., & Long, J. S. (1987). Parameterizing Non-linear Constraints in
Models with Latent Variables. Unpublished manuscript, Indiana University,
Department of Sociology, Bloomington, IN.
Yang-Wallentin, F., & Joreskog, K. G. (2001). Robust standard errors and chisquares in interaction models. In G. Marcoulides, & R. E. Schumacker (Eds.),
New developments and techniques in structural equation modeling (pp. 159–171).
Mahwah, NJ: Lawrence Erlbaum.

Y102005.indb 371

3/22/10 3:27:01 PM

17
Matrix Approach to Structural
Equation Modeling

Key Concepts
Eight matrices in SEM models
Matrix notation: measurement and structural models
Free, fixed, and constrained parameters
Structured means
Mean matrices: tau and kappa

17.1 General Overview of Matrix Notation
We have deliberately delayed presenting the matrix notation used in calculating structural equation models because we wanted to first present the
basic concepts, principles, and applications of SEM. SEM models are typically analyzed using the eight different matrices illustrated in Figure 17.1
(Hayduk, 1987); although a few new ones have emerged, for example, tau
and kappa. SEM models may use some combination of these matrices, but
not use all of the matrices in a given analysis, for example, path analysis
or confirmatory factor analysis.
In this chapter we consider the technical matrix notation associated
with the LISREL matrix command language. As described in Jöreskog
and Sörbom (1996), the structural model is written in terms of the following matrix equation:
h = Bh + Γx + z

(17.1)

373

Y102005.indb 373

3/22/10 3:27:01 PM

374

A Beginner’s Guide to Structural Equation Modeling

A vector of
endogenous
concepts
1

0
=

ηm
(m × 1)

+
0

(m × 1)

η1

A vector of
“errors” in the
conceptual model.
The covariances
among these errors
constitute Ψ

A matrix of
structural
coeﬃcients

(n × n)

Λy η

(p × m) (m × 1)

ζm

The vector
A matrix of
structural of endogenous
concepts
coeﬃcients

(p × 1)

The vector
of exogenous
concepts
+

Λx

δ
δ1

xq
(q × 1)

(m × n) (n × 1)

ηm

A vector of
observed
exogenous
indicators
x
x1

ηm

yp
(p × 1)

ζ
ζ1

η1

(m × m) (m × 1)

A vector of
observed
endogenous
indicators
y

The covariances
among those exogenous
concepts constitute Φ

A vector of
exogenous
concepts

βη

η1

Matrices of
structural
coeﬃcients

(q × n)(n × 1)

δq
(q × 1)

A vector of
“errors” in the
measurement model.
The covariances
among these errors
constitute Θ

(m × m)

(p × p)
A vector of
“errors” in the
measurement model.
The covariances
among these errors
constitute Θδ
(q × q)

Figure 17.1
Summary of the general structural equation model. (From Hayduk, L. A. (1987). Structural
equation modeling with LISREL: Essentials and advances. Baltimore, MD: Johns Hopkins
University Press.)

The latent dependent variables are denoted by h (eta) as a vector (m × 1)
of m such variables. The latent independent variables are denoted by x
(ksi) as a vector (n × 1) of n such variables. A matrix Φ (capital phi) contains the variances and covariance terms among these latent independent
variables. The relationships among the latent variables are denoted by B

Y102005.indb 374

3/22/10 3:27:02 PM

Matrix Approach to Structural Equation Modeling

375

(capital beta) and Γ (capital gamma), the elements of which are denoted by
[b] (lowercase beta) and [g] (lowercase gamma), respectively. The matrix B
is a m × m matrix of structure coefficients that relate the latent dependent
variables to one another. G is a m × n matrix of structure coefficients that
relate the latent independent variables to the latent dependent variables.
The error term z (zeta) in the structural model equation is a vector that
contains the equation prediction errors or disturbance terms. The matrix
Ψ (capital psi) contains the variances and covariance terms among these
latent dependent prediction equation errors.
As described in Jöreskog and Sörbom (1996), the measurement models
are written in the following set of matrix equations:
Y = Λy h + e,

(17.2)

for the latent dependent variables, and
X = Λx x + d,

(17.3)

for the latent independent variables. The observed variables are denoted
by the vector Y (p × 1) for the measures of the latent dependent variables
h (m × 1), and by the vector X (q × 1) for the measures of the latent independent variables x (n × 1). The relationships between the observed variables and the latent variables (typically referred to as factor loadings) are
denoted by the (p × m) matrix Λy (capital lambda sub y) for the Y’s, the
elements of which are denoted by [λy] (lowercase lambda sub y), and by
the q × n matrix Λx (capital lambda sub x) for the X’s, the elements of which
are denoted by [λx] (lowercase lambda sub x). Finally, the measurement
errors for the Y’s are denoted by the p × 1 vector e (lowercase epsilon)
and for the X’s by the q × 1 vector d (lowercase delta). The theta–epsilon
matrix Θe contains the variances and covariance terms among the errors
for the observed dependent variables. The theta–delta matrix Θd contains
the variances and covariance terms among the errors for the observed
independent variables.
The summary of the general structural equation model in matrix format
depicted by Hayduk (1987) should be studied in great detail. The three
equations diagrammed in matrix format correspond to the structural
model in Equation 17.1, the measurement model for the Y dependent variables in Equation 17.2, and the measurement model for the X independent
variables in Equation 17.3.
Obviously, not all of the matrices are used in every SEM model. We use our
examples from chapters 9 and 10 to illustrate the matrix notation for a structural equation model. In our first example in chapter 9 (see Figure 9.1), there
were two structure coefficients of interest. The first involved the influence of
Intelligence on Achievement1. The structure coefficient for this influence resides

Y102005.indb 375

3/22/10 3:27:02 PM

376

A Beginner’s Guide to Structural Equation Modeling

in the matrix Γ because it represents the relationship between the latent independent variable Intelligence and the latent dependent variable Achievement1.
The second structure coefficient involved the influence of Achievement1 on
Achievement2. This coefficient resides in the matrix B because it represents
the relationship between the latent dependent variable Achievement1 and
the latent dependent variable Achievement2. The final term in the structural
model of Equation 17.1 is z (zeta), an m × 1 vector of m equation errors or disturbances, which represents that portion of each latent dependent variable
that is not explained or predicted by the model.
In LISREL notation our structural equations are written as

η1 = γ 11ξ1 + ζ 1 ,
and

η2 = β21η1 + ζ 2 ,
respectively, or in the complete matrix equation as
η1   0
 =
η2   β21
 

ζ 1 
0  η1  γ 11 
   +   ξ1  +   ,
0  η   0 
ζ 2 
 2  
 

where the subscripts on b represent the rows for a latent dependent variable being predicted and columns for a latent dependent variable as the
predictor, respectively. The subscripts for g represent the rows for a latent
dependent variable being predicted and columns for a latent independent
variable as the predictor, respectively.
The values of 0 shown in the matrix equations for B and Γ represent
structure coefficients that we hypothesize to be equal to 0. For example,
because we did not specify that Intelligence influenced Achievement2, rather
than estimate g 21, we set that value to 0. Likewise, we did not specify that
Achievement2 influenced Achievement1, so we set b12 to 0. Finally, notice that
the diagonal values of B are also 0, that is, b11 and b 22. The diagonal values of B are always set to 0 because they indicate the extent to which a
latent dependent variable influences itself. These influences are never of
interest to the SEM researcher. In summary, our matrix equation suggests
that there are potentially four structure coefficients of interest, b12, b 21, g 11,
and g 21; however, our model includes only two of these coefficients. Other
structural models of these same latent variables can be developed that
contain different configurations of structure coefficients.
We now need to provide a more explicit definition of the measurement
models in our example. We have two different measurement models in

Y102005.indb 376

3/22/10 3:27:03 PM

Matrix Approach to Structural Equation Modeling

377

our example, one for the latent dependent variables and one for the latent
independent variables. In LISREL matrix notation these equations are
written for the Ys as
y1 = λ y11η1 + ε 1
y 2 = λ y21η1 + ε 2
y 3 = λ y32 η2 + ε 3
y 4 = λ y42 η2 + ε 4
and for the Xs as
x1 = λ x11ξ1 + δ 1
x2 = λ x21ξ1 + δ 2
The factor loadings and error terms also appear in their respective error
variance–covariance matrices. The complete matrix equation for the Ys is
written as
 y1   λ
   y11
 y2   λ
  =  y21
 y  0
 3 
   0
 y4 


 ε1 

 


0  η1
ε2
   +  
λ y32  η  ε 3
 2  

ε 4 
λ y42 

0

and for the Xs as
 x1   λ x11 
δ 
 ξ 1  +  1 
 =
 x2   λ x21 
δ 2 
where the subscripts in λy represent the rows for an observed Y variable
and the columns for a latent dependent variable, and those in λx represent
the rows for an observed X variable and the columns for a latent independent variable, respectively.

Y102005.indb 377

3/22/10 3:27:05 PM

378

A Beginner’s Guide to Structural Equation Modeling

The values of 0 shown in the matrix equations for Λy (and theoretically
for Λx, although not for this particular model) represent factor loadings
that we hypothesize to be equal to 0. For example, because we did not
specify that California1 was an indicator of Achievement2, rather than estimate λy12, we set that value to 0. Likewise, we specified that λy22, λy31, and
λy41 were set to 0.
There are several covariance terms that we need to define. From the
structural model, there are two covariance terms to consider. First, we
define Φ (capital phi) as an n × n covariance matrix of the n latent independent variables, the elements of which are denoted by [f] (lowercase phi).
The diagonal elements of Φ contain the variances of the latent independent variables. In our example, model Φ contains only one element, the
variance of Intelligence (denoted by f 11).
Second, let us define Ψ (psi) as an m × m covariance matrix of the m
equation errors z, the elements of which are denoted by [ψ] (lowercase
psi). The diagonal elements of Ψ contain the variances of the equation
errors—that is, the amount of unexplained variance for each equation. In
our example model Ψ contains two diagonal elements, one for each equation (denoted by ψ11 and ψ22).
From the measurement model there are two additional covariance
terms to be concerned with. First, we define Θe (capital theta sub epsilon)
as a p × p covariance matrix of the measurement errors for the Ys—that is,
e, the elements of which are denoted by (θe), lowercase theta sub epsilon.
The diagonal elements of Θe contain the variances of the measurement
errors for the Ys. In our example model Θe contains four diagonal elements, one for each Y. Second, let us define Θd (capital theta sub delta) as
a q × q covariance matrix of the measurement errors for the Xs—that is, d,
the elements of which are denoted by (θd), lowercase theta sub delta. The
diagonal elements of Θd contain the variances of the measurement errors
for the Xs. In our example model, Θd contains two diagonal elements, one
for each X.
There is one more covariance term that we need to define, and it represents the ultimate covariance term. To this point we have defined the
following eight different matrices: B, Γ, Λy, Λx, Φ, Ψ, Θd and Θe . From these
matrices we can generate an ultimate matrix of covariance terms that the
overall model implies, and this matrix is denoted by Σ (sigma). Officially,
Σ is a supermatrix composed of four submatrices, as follows:
 Σ yy

 Σ xy

Σ yx 

Σ xx 

(17.4)

This supermatrix certainly looks imposing, but it can be easily understood. First consider the submatrix in the upper left portion of Σ. It deals

Y102005.indb 378

3/22/10 3:27:05 PM

Matrix Approach to Structural Equation Modeling

379

with the covariance terms among the Ys, and in terms of our model can
be written as
Σ yy = [ Λ y [(Ι − Β)−1 (ΓΦΓ ′ + Ψ )(Ι − Β′)−1 ] Λ ′y + Θ ε ],

(17.5)

where I is an m × m identity matrix (i.e., a matrix having 1s on the diagonal and 0s on the off-diagonal). You can see in Equation 17.5 that all of the
matrices are involved except for those of the measurement model in the
X’s. That is, Equation 17.5 contains the matrices for the structural model
and for the measurement model in the Ys.
Consider next the submatrix in the lower right portion of Σ. It deals
with the covariance terms among the Xs and in terms of our model can
be written as
Σ xx = [ Λ x ΦΛ ′x + Θδ ]

(17.6)

As shown in Equation 17.6, the only matrices included are those that
involve the X side of the model. This particular portion of the model is the
same as the common factor analysis model, which you may recognize.
Finally, consider the submatrix in the lower left portion of Σ. It deals
with the covariance terms between the X’s and the Y’s and in terms of our
model can be written as
Σ xy = [ Λ x ΦΓ ′(Ι − B′)−1 Λ ′y ].

(17.7)

As shown in Equation 17.7, this portion of the model includes all of our
matrices except for the error terms, that is, Ψ, Θd and Θe . The submatrix in
the upper right portion of Σ is the transposed version of Equation 17.7 (i.e.,
the matrix of Equation 17.7 with rows and columns switched), so we need
not concern ourselves with it.

17.2 Free, Fixed, and Constrained Parameters
Let us return for a moment to our eight structural equation matrices B, Γ,
Λy, Λx, Φ, Ψ, Θd and Θe . In the structural model there are structure coefficients in matrices B and Γ. The covariance terms among structural equation
errors are in the matrix Ψ. In the measurement models for latent independent and dependent variables, there are factor loadings in the matrices Λx
and Λy, respectively, for their indicator variables. The covariance terms of
measurement errors for the latent independent and dependent variables
are in the matrices Θd and Θe, respectively. The covariance terms among

Y102005.indb 379

3/22/10 3:27:06 PM

380

A Beginner’s Guide to Structural Equation Modeling

the latent independent variables are in the matrix Φ. Each and every element in these eight matrices, if used in a particular model, must be specified to be a free parameter, a fixed parameter, or a constrained parameter.
A free parameter is a parameter that is unknown and one that you wish
to estimate. A fixed parameter is a parameter that is not free but rather is
fixed to a specified value, typically either 0 or 1. A constrained parameter
is a parameter that is unknown, but is constrained to be equal to one or
more other parameters.
For example, consider the following matrix B:
 0
B=
 β21

β12 

0 

The bs represent values in B that might be parameters of interest and thus
constitute free parameters. The 0s represent values in B that are fixed or
constrained to be equal to 0. These diagonal values of B represent the
influence of a latent dependent variable on itself, and by definition are
always fixed to 0. If our hypothesized model included only b 21, then b12
would also be fixed to 0. For the model specified in Figure 10.2 in chapter
10, B takes the following form:
 0
B=
 β21

0
.
0

For another example, consider the following matrix Λy with the factor
loadings for the latent dependent variable measurement model:
 λy
 11
 λy
Λ y =  21
 λ y31

 λ y41

λ y12 

λ y22 

λ y32 

λ y42 


Here the λy represent the values in Λy that might be parameters of interest
and would constitute free parameters. This specifies that we are allowing all of the parameters in Λy to be free so that each of our four indicator
variables (the Y’s) loads on each of our two latent dependent variables (the
h’s). However, in order to solve the identification problem for Λy, some
constraints are usually placed on this matrix whereby some of the parameters are fixed. We might specify that the first two indicator variables are
allowed only to load on the first latent dependent variable (h1) and the
latter two indicators on the second latent dependent variable (h2). Then,

Y102005.indb 380

3/22/10 3:27:07 PM

Matrix Approach to Structural Equation Modeling

381

Λy appears as
 λy
 11
 λy
Λ y =  21
0

 0



0 

λ y32 

λ y42 

0

Additional constraints in Λy may also be necessary for identification
purposes.
For the structural equation model in chapter 10 (Figure 10.2), the following structural equations are specified:
Aspirations = home background + ability + error
Achievement = aspirations + home background + ability + error
The matrix equation would be h = Bh + Γx + z and the elements of the
matrices are
 η1   0
 =
η2   β21

0   η1  γ 11
 + 
0  η2  γ 21

γ 12   ξ1  ζ 1 
   +  .
γ 22  ξ2  ζ 2 

The matrix equation for the latent dependent variable measurement model
is Y = Λy h + e, and the elements of the matrices are
 y1 
  1
 y2   λ
  =  y21
 y  0
 3 
   0
 y4 


ε 
   1
η
  1  ε 2 
.
+
1  η   ε 3 
 2   
λ y42 
ε 4 

0
0

The matrix equation for the latent independent variable measurement
model is X = Λx x + d and the elements of the matrices are
 x1 
  1
 x2   λ x
   21
 x  =  λ x31
 3 
  0
 x4  
   0
 x5 

Y102005.indb 381


δ 1 

 
δ

ξ   2 
0   1  + δ 3  .

   
1  ξ2  δ 4 

 
δ 5 
λ x52 

0
0

3/22/10 3:27:08 PM

382

A Beginner’s Guide to Structural Equation Modeling

Recall that for each dependent and independent latent variable we fixed
one factor loading of an observed variable to 1. This was necessary to
identify the model and to fix the scale for the latent variables.
The covariance terms are written next. The covariance matrix for the
latent independent variables is
φ
Φ =  11
φ21


.
φ22 

The covariance matrix for the structural equation errors is
ψ
Ψ =  11
ψ 21


.
ψ 22 

The covariance matrices for the measurement errors are written as follows, first for the indicators of the latent independent variables by
θδ
 11
 0

Θδ =  0

 0
 0


θδ 22
θδ 32

θδ 33

θδ 44








θδ 55 


and second, for the indicators of the latent dependent variables by
θε
 11
 0
Θε = 
 0

 0

θε22
0

θε33







θε 44 


Note : This matrix output is possible by including the LISREL OUTPUT
command in the LISREL–SIMPLIS program for the model in chapter 10.

17.3 LISREL Model Example in Matrix Notation
The LISREL matrix command language program works directly from
the matrix notation previously discussed and is presented here for the
example in chapter 10. The basic LISREL matrix command language program includes TITLE, DATA (DA), INPUT, MODEL (MO), and OUTPUT

Y102005.indb 382

3/22/10 3:27:09 PM

Matrix Approach to Structural Equation Modeling

383

(OU) program statements. The TITLE lines are optional. The user’s guide
provides an excellent overview of the various commands and their purpose (Jöreskog & Sörbom, 1996). The DA statement identifies the number
of input variables in the variance–covariance matrix, the NO statement
indicates the number of observations, and MA identifies the kind of
matrix to be analyzed, not the kind of matrix to be inputted: MA = CM,
covariance matrix; MA = KM, correlation matrix based on raw scores or
normal scores; MA = MM, matrix of moments (means) about zero; MA =
AM, augmented moment matrix; MA = OM, special correlation matrix of
optimal scores from PRELIS2; and MA = PM, correlation matrix of polychoric (ordinal variables) or polyserial (ordinal and continuous variables)
correlations. The SE statement must be used to select and/or reorder variables used in the analysis of a model (the Y variables must be listed first).
An external raw score data file can be read using the RA statement with
the FI and FO subcommands, for example, RA FI = raw.dat FO. The FO
subcommand permits the specification of how observations are to be read
(Note: for fixed, a FORMAT statement must be enclosed in parentheses;
for free-field, an asterisk is placed in the first column, which appears on
the line following the RA command). If FI or UN (logical unit number of
a FORTRAN file) subcommands are not used, then the data must directly
follow the RA command and be included in the program.
In the following LISREL matrix command language program, a lower
diagonal variance–covariance matrix is input, hence, the use of the CM
statement. The SY subcommand, which reads only the lower diagonal elements of a matrix, has been omitted because it is the default option for
matrix input. The LA statement provides for up to eight characters for
variable labels, with similar subcommand options for input and specifications as with the RA command for data input. (Note: A lower case c permits line continuation for various commands). The LE command permits
variable labels for the latent dependent variables, and the LK command
permits variable labels for the latent independent variables.
The MO command specifies the model for LISREL analysis. The subcommands specify the number of Y variables (ny), number of X variables
(nx), number of latent dependent variables (ne), and number of latent independent variables (nk). The form and mode of the eight LISREL parameter matrices must be specified and are further explained in the user’s
guide (Jöreskog & Sörbom, 1996). The FU parameter indicates a full
non-symmetric matrix form, and FI indicates a fixed matrix mode,
in contrast to a free mode (FR). The DI statement indicates a diagonal
matrix form, and the SY statement indicates a symmetric matrix form. It is
strongly recommended that any designation of a LISREL model for analysis include the presentation of the eight matrices in matrix form. This will
greatly ease the writing of the MO command and the identification of fixed
or free parameters in the matrices on the FR and VA commands. The VA

Y102005.indb 383

3/22/10 3:27:09 PM

384

A Beginner’s Guide to Structural Equation Modeling

command assigns numerical values to the fixed parameters. The OU command permits the selection of various output procedures. One feature of
interest on the OU command is the AM option, which provides for automatic model specification by freeing at each step the fixed or constrained
parameters with the largest modification indices, although, as previously
noted, this should not be the sole criterion for model modification.
The LISREL matrix command language program used to analyze the
model in Figure 10.2 of chapter 10, using the default maximum likelihood
estimation method, is as follows:
Modified Model in Figure 10.2, Chapter 10
da ni=9 no=200 ma=cm
cm sy
1.024
.792 1.077
1.027 .919 1.844
.756 .697 1.244 1.286
.567 .537 .876 .632 .852
.445 .424 .677 .526 .518 .670
.434 .389 .635 .498 .475 .545 .716
.580 .564 .893 .716 .546 .422 .373 .851
.491 .499 .888 .646 .508 .389 .339 .629 .871
la
EDASP OCASP VERBACH QUANTACH FAMINC FAED MOED VERBAB
c QUANTAB mo ny=4 nx=5 ne=2 nk=2 be=fu,fi ga=fu,fi ph=sy,fi ps=di,fi
c ly=fu,fi lx=fu,fi td=fu,fi te=fu,fi
le
aspire achieve
lk
home ability
fr be(2,1) ga(1,1) ga(1,2) ga(2,1) ga(2,2)
c ly(2,1) ly(4,2) lx(2,1) lx(3,1) lx(5,2)
c te(1,1) te(2,2) te(3,3) te(4,4) td(1,1) td(2,2) td(3,3)
c td(4,4) td(5,5)
c ps(1,1) ps(2,2) ph(1,1) ph(2,2) ph(2,1) td(3,2)
va 1.0 ly(1,1) ly(3,2) lx(1,1) lx(4,2)
ou me=ml all
(Note : The c values in the LISREL program denote line continuations in
program statements.)

The LISREL matrix command language requires the user to specifically
understand the nature, form, and mode of the eight matrices, and thereby
fully comprehend the model being specified for analysis, even though all
eight matrices may not be used in a particular SEM model. We present the
LISREL output from this program, but do so in an edited and condensed
format. We challenge you to find the various matrices we have described
in this chapter in the computer output.

Y102005.indb 384

3/22/10 3:27:09 PM

385

Matrix Approach to Structural Equation Modeling

LISREL8 Matrix Program Output (Edited and Condensed)
Modified Model in Figure 10.2, chapter 10

Number
Number
Number
Number
Number
Number

of
of
of
of
of
of

Input Variables 9
Y - Variables
4
X - Variables
5
ETA - Variables 2
KSI - Variables 2
Observations 200

Covariance Matrix
EDASP
OCASP
VERBACH QUANTACH FAMINC
FAED
-------- -------- -------- -------- -------- -------EDASP 1.02
OCASP 0.79
1.08
VERBACH 1.03
0.92
1.84
QUANTACH 0.76
0.70
1.24
1.29
FAMINC 0.57
0.54
0.88
0.63
0.85
FAED 0.45
0.42
0.68
0.53
0.52
0.67
MOED 0.43
0.39
0.64
0.50
0.47
0.55
VERBAB 0.58
0.56
0.89
0.72
0.55
0.42
QUANTAB 0.49
0.50
0.89
0.65
0.51
0.39
Covariance Matrix

MOED
VERBAB
QUANTAB

MOED
-------0.72
0.37
0.34

VERBAB
--------

QUANTAB
--------

0.85
0.63

0.87

LISREL Estimates (Maximum Likelihood)
LAMBDA-Y

EDASP
OCASP

VERBACH
QUANTACH

aspire
-------1.00
0.92
(0.06)
14.34
- - -

achieve
-------- - -

1.00
0.75
(0.04)
18.13

LAMBDA-X

FAMINC

Y102005.indb 385

home
-------1.00

ability
-------- -

3/22/10 3:27:10 PM

386

A Beginner’s Guide to Structural Equation Modeling

FAED
MOED
VERBAB
QUANTAB

0.78
(0.06)
12.18
0.72
(0.07)
10.37
- - -

- - 1.00
0.95
(0.07)
14.10

BETA

aspire
achieve

aspire
-------- 0.53
(0.12)
4.56

achieve
-------- - -

home
-------0.51
(0.15)
3.29
0.30
(0.16)
1.87

ability
-------0.45
(0.15)
2.96
0.69
(0.16)
4.27

GAMMA

aspire
achieve

Covariance Matrix of ETA and KSI

aspire
achieve
home
ability

aspire
-------0.86
1.02
0.57
0.57

achieve
--------

home
--------

ability
--------

1.65
0.87
0.91

0.66
0.54

0.66

PHI
home
-------home

ability

Y102005.indb 386

0.54
(0.07)
7.64

ability
-------0.66
(0.09)
7.32
0.66
(0.09)
7.51

3/22/10 3:27:10 PM

387

Matrix Approach to Structural Equation Modeling

PSI
Note: This matrix is diagonal.
aspire
-------0.32
(0.06)
5.61

achieve
-------0.23
(0.06)
3.97

Squared Multiple Correlations for Structural Equations
aspire
-------0.63

achieve
-------0.86

Squared Multiple Correlations for Reduced Form
aspire
-------0.63
THETA-EPS
EDASP
OCASP
--------------0.16
0.35
(0.04)
(0.05)
3.88
7.36

achieve
-------0.81
VERBACH
-------0.19
(0.05)
3.81

QUANTACH
-------0.35
(0.04)
7.95

Squared Multiple Correlations for Y - Variables
EDASP
-------0.84

OCASP
-------0.67

VERBACH
-------0.90

QUANTACH
-------0.73

THETA-DELTA

FAED

FAMINC
-------0.19
(0.04)
4.74
- -

MOED

- -

VERBAB

- -

0.27
(0.03)
7.66
0.17
(0.03)
5.28
- -

QUANTAB

- -

FAMINC

Y102005.indb 387

FAED
--------

MOED
--------

0.37
(0.04)
8.50
- - -

VERBAB
--------

0.19
(0.03)
5.41
- -

QUANTAB
--------

0.27
(0.04)
7.20

3/22/10 3:27:10 PM

388

A Beginner’s Guide to Structural Equation Modeling

Squared Multiple Correlations for X - Variables
FAMINC
-------0.78

FAED
-------0.60

MOED
-------0.48

VERBAB
-------0.78

QUANTAB
-------0.69

Goodness-of-Fit Statistics
Degrees of Freedom = 20
Minimum Fit Function Chi-Square = 19.17 (P = 0.51)
Normal Theory Weighted Least Squares Chi-Square = 18.60
(P = 0.55)
Estimated Non-centrality Parameter (NCP) = 0.0
90 Percent Confidence Interval for NCP = (0.0 ; 12.67)
Minimum Fit Function Value = 0.096
Population Discrepancy Function Value (F0) = 0.0
90 Percent Confidence Interval for F0 = (0.0 ; 0.064)
Root Mean Square Error of Approximation (RMSEA) = 0.0
90 Percent Confidence Interval for RMSEA = (0.0 ; 0.056)
P-Value for Test of Close Fit (RMSEA < 0.05) = 0.91
Expected Cross-Validation Index (ECVI) = 0.35
90 Percent Confidence Interval for ECVI = (0.35 ; 0.42)
ECVI for Saturated Model = 0.45
ECVI for Independence Model = 13.72
Chi-Square for Independence Model with 36 Degrees of
Freedom = 2712.06
Independence AIC = 2730.06
Model AIC = 68.60
Saturated AIC = 90.00
Independence CAIC = 2768.74
Model CAIC = 176.05
Saturated CAIC = 283.42
Normed Fit Index (NFI) = 0.99
Non-Normed Fit Index (NNFI) = 1.00
Parsimony Normed Fit Index (PNFI) = 0.55
Comparative Fit Index (CFI) = 1.00
Incremental Fit Index (IFI) = 1.00
Relative Fit Index (RFI) = 0.99
Critical N (CN) = 391.00
Root Mean Square Residual (RMR) = 0.015
Standardized RMR = 0.015
Goodness-of-Fit Index (GFI) = 0.98
Adjusted Goodness-of-Fit Index (AGFI) = 0.95
Parsimony Goodness-of-Fit Index (PGFI) = 0.44

Y102005.indb 388

3/22/10 3:27:10 PM

389

Matrix Approach to Structural Equation Modeling

Fitted Covariance Matrix

EDASP
OCASP
VERBACH
QUANTACH
FAMINC
FAED
MOED
VERBAB
QUANTAB

EDASP
OCASP
VERBACH QUANTACH FAMINC
FAED
-------- -------- -------- -------- -------- -------1.02
0.79
1.08
1.02
0.93
1.84
0.77
0.70
1.24
1.29
0.57
0.53
0.87
0.66
0.85
0.45
0.41
0.68
0.51
0.52
0.67
0.41
0.38
0.63
0.47
0.48
0.54
0.57
0.52
0.91
0.69
0.54
0.42
0.54
0.49
0.87
0.65
0.51
0.40

Fitted Covariance Matrix

MOED
VERBAB
QUANTAB

MOED
-------0.72
0.39
0.37

VERBAB
--------

QUANTAB
--------

0.85
0.63

0.87

Fitted Residuals

EDASP
OCASP
VERBACH
QUANTACH
FAMINC
FAED
MOED
VERBAB
QUANTAB

EDASP
OCASP
VERBACH QUANTACH FAMINC
FAED
-------- -------- -------- -------- -------- -------0.00
0.00
0.00
0.01
-0.01
0.00
-0.01
-0.01
0.00
0.00
-0.01
0.01
0.01
-0.02
0.00
0.00
0.01
0.00
0.01
0.00
0.00
0.02
0.01
0.01
0.03
0.00
0.00
0.01
0.04
-0.02
0.03
0.01
0.00
-0.05
0.00
0.02
-0.01
0.00
-0.01

Fitted Residuals

MOED
VERBAB
QUANTAB

MOED
-------0.00
-0.01
-0.03

VERBAB
--------

QUANTAB
--------

0.00
0.00

0.00

Summary Statistics for Fitted Residuals
Smallest Fitted Residual = -0.05
Median Fitted Residual = 0.00
Largest Fitted Residual = 0.04

Y102005.indb 389

3/22/10 3:27:10 PM

390

A Beginner’s Guide to Structural Equation Modeling

Stemleaf Plot
- 4|8
- 3|
- 2|842
- 1|4400
- 0|886542100000000000000
0|2469999
1|1123
2|0067
3|
4|3
Standardized Residuals

EDASP
OCASP
VERBACH
QUANTACH
FAMINC
FAED
MOED
VERBAB
QUANTAB

EDASP
OCASP
VERBACH QUANTACH FAMINC
FAED
-------- -------- -------- -------- -------- -------- - - 1.26
-1.01
- -0.52
-0.23
- - -0.64
0.45
0.55
-1.17
- -0.25
0.45
-0.23
0.58
0.15
- 0.82
0.30
0.36
0.91
-0.15
- 0.88
1.93
-2.34
1.50
0.72
0.10
-2.53
0.16
1.59
-0.38
-0.13
-0.50

Standardized Residuals

MOED
VERBAB
QUANTAB

MOED
-------- -0.63
-1.10

VERBAB
--------

QUANTAB
--------

- - -

- -

Summary Statistics for Standardized Residuals
Smallest Standardized Residual = -2.53
Median Standardized Residual = 0.00
Largest Standardized Residual = 1.93
Stemleaf Plot
-

Y102005.indb 390

2|5
2|3
1|
1|210
0|6655
0|4322210000000000000
0|122344
0|5567899
1|3
1|569

3/22/10 3:27:11 PM

Matrix Approach to Structural Equation Modeling

391

Modification Indices and Expected Change
Modification Indices for LAMBDA-Y

EDASP
OCASP
VERBACH
QUANTACH

aspire
-------- - 0.32
0.32

achieve
-------0.30
0.30
- - -

Expected Change for LAMBDA-Y

EDASP
OCASP
VERBACH
QUANTACH

aspire
-------- - 0.12
-0.09

achieve
-------0.28
-0.26
- - -

Standardized Expected Change for LAMBDA-Y

EDASP
OCASP
VERBACH
QUANTACH

aspire
-------- - 0.11
-0.09

achieve
-------0.36
-0.33
- - -

Modification Indices for LAMBDA-X

FAMINC
FAED
MOED
VERBAB
QUANTAB

home
-------- - - 0.63
0.63

ability
-------0.40
0.11
0.49
- - -

Expected Change for LAMBDA-X

FAMINC
FAED
MOED
VERBAB
QUANTAB

Y102005.indb 391

home
-------- - - 0.16
-0.16

ability
-------0.18
0.04
-0.08
- - -

3/22/10 3:27:11 PM

392

A Beginner’s Guide to Structural Equation Modeling

Standardized Expected Change for LAMBDA-X
home
-------- - - 0.13
-0.13

FAMINC
FAED
MOED
VERBAB
QUANTAB
No
No
No
No

Non-Zero
Non-Zero
Non-Zero
Non-Zero

Modification
Modification
Modification
Modification

ability
-------0.15
0.03
-0.06
- - -

Indices
Indices
Indices
Indices

for
for
for
for

BETA
GAMMA
PHI
PSI

Modification Indices for THETA-EPS

EDASP
OCASP
VERBACH
QUANTACH

EDASP
-------- - 2.32
0.17

OCASP
--------

VERBACH
--------

- 1.91
0.01

QUANTACH
--------

- - -

- -

Expected Change for THETA-EPS

EDASP
OCASP
VERBACH
QUANTACH

EDASP
-------- - 0.05
-0.01

OCASP
--------

VERBACH
--------

- -0.05
0.00

QUANTACH
--------

- - -

- -

Modification Indices for THETA-DELTA-EPS

FAMINC
FAED
MOED
VERBAB
QUANTAB

EDASP
-------0.12
0.62
1.13
0.51
4.92

OCASP
-------0.06
0.32
0.40
1.13
0.30

VERBACH
-------0.86
0.30
0.02
8.44
5.47

QUANTACH
-------2.09
0.15
0.37
3.03
0.94

Expected Change for THETA-DELTA-EPS

FAMINC
FAED
MOED

Y102005.indb 392

EDASP
--------0.01
-0.01
0.02

OCASP
-------0.01
0.01
-0.02

VERBACH
-------0.03
-0.01
0.00

QUANTACH
--------0.04
0.01
0.01

3/22/10 3:27:11 PM

393

Matrix Approach to Structural Equation Modeling

VERBAB
QUANTAB

0.02
-0.06

0.03
0.02

-0.09
0.07

0.05
-0.03

Modification Indices for THETA-DELTA

FAMINC
FAED
MOED
VERBAB
QUANTAB

FAMINC
-------- 0.02
0.02
0.15
0.02

FAED
--------

MOED
--------

VERBAB
--------

QUANTAB
--------

- - 0.14
0.05

- 0.36
0.59

- - -

- -

Expected Change for THETA-DELTA

FAMINC
FAED
MOED
VERBAB
QUANTAB

FAMINC
-------- 0.00
0.00
0.01
0.00

FAED
--------

MOED
--------

VERBAB
--------

QUANTAB
--------

- - 0.01
0.00

- -0.01
-0.02

- - -

- -

Maximum Modification Index is 8.44 for Element (4, 3)
of THETA DELTA-EPSILON
Covariances
Y - ETA

aspire
achieve

EDASP
-------0.86
1.02

OCASP
-------0.79
0.93

VERBACH
-------1.02
1.65

QUANTACH
-------0.77
1.24

EDASP
-------0.57
0.57

OCASP
-------0.53
0.52

VERBACH
-------0.87
0.91

QUANTACH
-------0.66
0.69

Y - KSI

home
ability
X - ETA

aspire
achieve

Y102005.indb 393

FAMINC
-------0.57
0.87

FAED
-------0.45
0.68

MOED
-------0.41
0.63

VERBAB
QUANTAB
-------- -------0.57
0.54
0.91
0.87

3/22/10 3:27:11 PM

394

A Beginner’s Guide to Structural Equation Modeling

X - KSI

home
ability

FAMINC
-------0.66
0.54

FAED
-------0.52
0.42

MOED
-------0.48
0.39

VERBAB
QUANTAB
-------- -------0.54
0.51
0.66
0.63

First Order Derivatives
LAMBDA-Y

EDASP
OCASP
VERBACH
QUANTACH

aspire
-------0.00
0.00
-0.01
0.02

achieve
--------0.01
0.01
0.00
0.00

home
-------0.00
0.00
0.00
-0.02
0.02

ability
--------0.01
-0.01
0.03
0.00
0.00

LAMBDA-X

FAMINC
FAED
MOED
VERBAB
QUANTAB
BETA

aspire
achieve

aspire
-------0.00
0.00

achieve
-------0.00
0.00

home
-------0.00
0.00

ability
-------0.00
0.00

home
-------0.00
0.00

ability
--------

GAMMA

aspire
achieve
PHI

home
ability

Y102005.indb 394

0.00

3/22/10 3:27:11 PM

395

Matrix Approach to Structural Equation Modeling

PSI

aspire
achieve

aspire
-------0.00
0.00

achieve
-------0.00

THETA-EPS

EDASP
OCASP
VERBACH
QUANTACH

EDASP
-------0.00
0.00
-0.24
0.07

OCASP
--------

VERBACH
--------

QUANTACH
--------

0.00
0.21
-0.01

0.00
0.00

0.00

OCASP
--------0.04
-0.13
0.13
-0.20
-0.10

VERBACH
--------0.15
0.13
-0.03
0.48
-0.38

QUANTACH
-------0.26
-0.09
-0.13
-0.32
0.17

THETA-DELTA-EPS

FAMINC
FAED
MOED
VERBAB
QUANTAB

EDASP
-------0.07
0.21
-0.26
-0.15
0.44

THETA-DELTA

FAMINC
FAED
MOED
VERBAB
QUANTAB

FAMINC
-------0.00
-0.03
0.03
-0.08
-0.03

FAED
--------

MOED
--------

0.00
0.00
-0.10
-0.06

0.00
0.15
0.18

VERBAB
QUANTAB
-------- --------

0.00
0.00

0.00

Factor Scores Regressions
ETA
EDASP
OCASP
VERBACH QUANTACH FAMINC
FAED
-------- -------- -------- -------- -------- -------aspire
0.50
0.21
0.10
0.04
0.04
0.02
achieve
0.12
0.05
0.52
0.22
0.07
0.03
ETA

aspire
achieve

Y102005.indb 395

MOED
-------0.01
0.01

VERBAB
-------0.02
0.11

QUANTAB
-------0.01
0.07

3/22/10 3:27:12 PM

396

A Beginner’s Guide to Structural Equation Modeling

KSI
EDASP
OCASP
VERBACH QUANTACH FAMINC
FAED
-------- -------- -------- -------- -------- -------home
0.05
0.02
0.07
0.03
0.41
0.19
ability
0.02
0.01
0.11
0.04
0.07
0.03
KSI

home
ability

MOED
-------0.06
0.01

VERBAB
-------0.07
0.37

QUANTAB
-------0.04
0.24

Standardized Solution
LAMBDA-Y

EDASP
OCASP
VERBACH
QUANTACH

aspire
-------0.93
0.85
- - -

achieve
-------- - 1.29
0.97

home
-------0.81
0.64
0.59
- - -

ability
-------- - - 0.81
0.77

LAMBDA-X

FAMINC
FAED
MOED
VERBAB
QUANTAB
BETA

aspire
achieve

aspire
-------- 0.38

achieve
-------- - -

GAMMA

aspire
achieve

Y102005.indb 396

home
-------0.44
0.19

ability
-------0.39
0.43

3/22/10 3:27:12 PM

397

Matrix Approach to Structural Equation Modeling

Correlation Matrix of ETA and KSI

aspire
achieve
home
ability

aspire
-------1.00
0.85
0.76
0.75

achieve
--------

home
--------

ability
--------

1.00
0.83
0.87

1.00
0.81

1.00

PSI
Note: This matrix is diagonal.
aspire
-------0.37

achieve
-------0.14

Regression Matrix ETA on KSI (Standardized)

aspire
achieve

home
-------0.44
0.36

ability
-------0.39
0.58

Total and Indirect Effects
Total Effects of KSI on ETA

aspire
achieve

home
-------0.51
(0.15)
3.29
0.57
(0.17)
3.26

ability
-------0.45
(0.15)
2.96
0.92
(0.18)
5.20

Indirect Effects of KSI on ETA

aspire
achieve

Y102005.indb 397

home
-------- 0.27
(0.10)
2.63

ability
-------- 0.23
(0.09)
2.62

3/22/10 3:27:12 PM

398

A Beginner’s Guide to Structural Equation Modeling

Total Effects of ETA on ETA

aspire
achieve

aspire
-------- 0.53
(0.12)
4.56

achieve
-------- - -

Largest Eigenvalue of B*B’ (Stability Index) is 0.276
Total Effects of ETA on Y

EDASP
OCASP
VERBACH
QUANTACH

aspire
-------1.00
0.92
(0.06)
14.34
0.53
(0.12)
4.56
0.40
(0.09)
4.48

achieve
-------- - 1.00
0.75
(0.04)
18.13

Indirect Effects of ETA on Y

EDASP
OCASP
VERBACH
QUANTACH

aspire
-------- - 0.53
(0.12)
4.56
0.40
(0.09)
4.48

achieve
-------- - - - -

Total Effects of KSI on Y

EDASP
OCASP

Y102005.indb 398

home
-------0.51
(0.15)
3.29
0.46
(0.14)
3.25

ability
-------0.45
(0.15)
2.96
0.41
(0.14)
2.93

3/22/10 3:27:12 PM

Matrix Approach to Structural Equation Modeling

VERBACH
QUANTACH

0.57
(0.17)
3.26
0.43
(0.13)
3.23

399

0.92
(0.18)
5.20
0.69
(0.14)
5.09

Standardized Total and Indirect Effects
Standardized Total Effects of KSI on ETA

aspire
achieve

home
-------0.44
0.36

ability
-------0.39
0.58

Standardized Indirect Effects of KSI on ETA

aspire
achieve

home
-------- 0.17

ability
-------- 0.15

Standardized Total Effects of ETA on ETA

aspire
achieve

aspire
-------- 0.38

achieve
-------- - -

Standardized Total Effects of ETA on Y

EDASP
OCASP
VERBACH
QUANTACH

aspire
-------0.93
0.85
0.49
0.37

achieve
-------- - 1.29
0.97

Standardized Indirect Effects of ETA on Y

EDASP
OCASP
VERBACH
QUANTACH

Y102005.indb 399

aspire
-------- - 0.49
0.37

achieve
-------- - - - -

3/22/10 3:27:13 PM

400

A Beginner’s Guide to Structural Equation Modeling

Standardized Total Effects of KSI on Y

EDASP
OCASP
VERBACH
QUANTACH

home
-------0.41
0.38
0.46
0.35

ability
-------0.36
0.33
0.75
0.56

At this point, we leave it up to the reader to extract the factor loadings,
error variances, structure coefficients, and disturbance terms from the
various matrices indicated in the standardized solution. It is also helpful to determine the direct and indirect effects indicated in the model.
The model-fit indices indicated that the data fit the modified theoretical
model.

17.4 Other Models in Matrix Notation
This section presents the matrix approach to the path model, the multiplesample model, the structured means model and two types of interaction
models in structural equation modeling. The reader is referred to the previous chapters and references in the book for further detail and explanation of these types of models.
17.4.1 Path Model
The path model in LISREL matrix notation is written as
Y = BY + ΓX + z,
and thus there is no measurement model. Of the eight LISREL matrices,
for the path model we only have the following: B, Γ, Φ, and Ψ.
As an example path model, we again consider the union sentiment
model as previously shown in Figure 7.1 of Chapter 7. The structural equations in terms of variable names are
Deference (Y1) = Age (X1) + error1
Support (Y2) = Age (X1) + Deference (Y1) + error2
Sentiment (Y3) = Years (X2) + Deference (Y1) + Support (Y2) + error3.

Y102005.indb 400

3/22/10 3:27:13 PM

Matrix Approach to Structural Equation Modeling

401

In terms of matrix equations, this translates into the structural equation matrices:
 Y1   0
  
Y2  =  β21
Y3   β31
  

0
0
β32

0   Y1  γ 11
  
0  Y2  + γ 21
0  Y3   0


  X 1  ζ 1 
  X  + ζ  .
γ 32   2   2 
0
0

Finally, the relevant LISREL matrices for this model are as follows:
 0

B =  β21
 β31


0
0
β32

0

0
0 

γ 11

Γ = γ 21
 0


0 

0 
γ 32 

φ
Φ =  11
φ21



φ22 

ψ 11

Ψ= 0
 0


ψ 22
0




ψ 33 

The LISREL path model program would therefore define these matrices
as follows:
Union Sentiment of Textile Workers
DA NI=5 NO=173 MA=CM
CM SY
14.610
-5.250 11.017
-8.057 11.087 31.971
-0.482
0.677 1.559 1.021
-18.857 17.861 28.250 7.139 215.662
LA
Defer Support Sentim Years Age
SE
1 2 3 5 4
MO NY=3 NX=2 BE=FU,FI GA=FU,FI PH=FU,FR PS=DI
FR BE(2,1) BE(3,1) BE(3,2) GA(1,1) GA(2,1) GA(3,2)
OU ND=2

Y102005.indb 401

3/22/10 3:27:14 PM

402

A Beginner’s Guide to Structural Equation Modeling

Selected computer output from the LISREL path model program would be:
Union Sentiment of Textile Workers
Number of Iterations = 8
LISREL Estimates (Maximum Likelihood)
BETA

Defer
Support
Sentim

Defer
-------- -0.28
(0.06)
-4.58
-0.22
(0.10)
-2.23

Support
-------- - -

Sentim
-------- - -

0.85
(0.11)
7.53

- -

GAMMA

Defer
Support
Sentim

Age
--------0.09
(0.02)
-4.65
0.06
(0.02)
3.59
- -

Years
-------- - 0.86
(0.34)
2.52

PHI

Age
Years

Age
-------215.66
(23.39)
9.22
7.14
(1.26)
5.65

Years
--------

1.02
(0.11)
9.22

PSI
Note: This matrix is diagonal.
Defer
-------12.96
(1.41)
9.22

Y102005.indb 402

Support
-------8.49
(0.92)
9.22

Sentim
-------19.45
(2.11)
9.22

3/22/10 3:27:14 PM

403

Matrix Approach to Structural Equation Modeling

Squared Multiple Correlations for Structural Equations
Defer
-------0.11

Support
-------0.23

Sentim
-------0.39

Goodness-of-Fit Statistics
Degrees of Freedom = 3
Minimum Fit Function Chi-Square = 1.25 (P = 0.74)
Normal Theory Weighted Least Squares Chi-Square = 1.25
(P = 0.74)
Estimated Non-centrality Parameter (NCP) = 0.0
90 Percent Confidence Interval for NCP = (0.0 ; 4.20)
Minimum Fit Function Value = 0.0073
Population Discrepancy Function Value (F0) = 0.0
90 Percent Confidence Interval for F0 = (0.0 ; 0.025)
Root Mean Square Error of Approximation (RMSEA) = 0.0
90 Percent Confidence Interval for RMSEA = (0.0 ; 0.091)
P-Value for Test of Close Fit (RMSEA < 0.05) = 0.84
Expected Cross-Validation Index (ECVI) = 0.16
90 Percent Confidence Interval for ECVI = (0.16 ; 0.18)
ECVI for Saturated Model = 0.18
ECVI for Independence Model = 1.46
Chi-Square for Independence Model with 10 Degrees of Freedom
= 238.10
Independence AIC = 248.10
Model AIC = 25.25
Saturated AIC = 30.00
Independence CAIC = 268.87
Model CAIC = 75.09
Saturated CAIC = 92.30
Normed Fit Index (NFI) = 0.99
Non-Normed Fit Index (NNFI) = 1.03
Parsimony Normed Fit Index (PNFI) = 0.30
Comparative Fit Index (CFI) = 1.00
Incremental Fit Index (IFI) = 1.01
Relative Fit Index (RFI) = 0.98
Critical N (CN) = 1560.66
Root Mean Square Residual (RMR) = 0.73
Standardized RMR = 0.015
Goodness-of-Fit Index (GFI) = 1.00
Adjusted Goodness-of-Fit Index (AGFI) = 0.99
Parsimony Goodness-of-Fit Index (PGFI) = 0.20

Y102005.indb 403

3/22/10 3:27:14 PM

404

A Beginner’s Guide to Structural Equation Modeling

17.4.2 Multiple-Sample Model
The multiple-sample model in LISREL matrix notation for the measurement model is written as
Y = Λ(g)
y h + e
for the latent dependent indicator variables, and
X = Λ(g)
x x + d
for the latent independent indicator variables, where g = 1 to G groups
and the other terms are as previously defined. The structural model can be
written as follows:
h = B(g) h + Γ(g) x + z
The four covariance matrices that you are already familiar with are written as: Φ(g), Ψ(g), Θd(g), and Θ(g)
e . The measurement and structural equations
yield parameter estimates for each of the eight matrices for each group,
B(g), Γ(g), Λy(g), Λx(g), Φ(g), Ψ(g), Θd(g) and Θe(g).
For instance, with two groups we may be interested in testing whether
the factor loadings are equivalent. These hypotheses for the latent dependent variables are written as
Λy(1) = Λy(2)
and for the latent independent variables as
Λx(1) = Λx(2)
One might also hypothesize that any of the other matrices are equivalent
so that
(2)
Lomax d corr. → Θ(1)
d = Θd
(2)
Schumacker d corr. → Θ(1)
e = Θe

B(1) = B(2)
Γ(1) = Γ(2)
Φ(1) = Φ(2)
Ψ(1) = Ψ(2)
Thus, the groups can be evaluated to determine which matrices are equivalent, and which are different.

Y102005.indb 404

3/22/10 3:27:14 PM

Matrix Approach to Structural Equation Modeling

405

17.4.3 Structured Means Model
The structured means model in LISREL matrix notation for the measurement model of the latent dependent indicator variables is written as
(g)
Y = t (g)
y + Λy h + e,

and for the latent independent indicator variables written as
(g)
X = t (g)
x + Λx x + d.

We denote t y and t x as vectors of constant intercept terms (means) for
the indicator variables, and the other terms are as previously defined
(Jöreskog & Sörbom, 1996) denoted these intercept terms as t; other publications have used u instead]. The structural model is now written as
h = a (g) + B(g) h + Γ (g) x + z,
where a is a vector of constant intercept terms (means) for the structural
equations and the other terms are as previously defined. In most SEM models the intercept terms are assumed to be zero, so the structured means
model is a special application of SEM used in the analysis of variance as
well as slope and intercept models. In the structured means model, the
intercept term is not zero and therefore estimated (see chapter 6 for intercept terms in regression using CONST term).
In addition to the means of indicator variables being estimated, other
latent variable means can be estimated. The mean of each latent independent variable x is given by k; for example, k 1 denotes the mean for x 1. The
mean of each latent dependent variable is given by (I – B)-1 (a + Γk).
In addition to the hypotheses given previously for the simple multiplesample model, the structured means model can also examine a, the group
effects for each structural equation, and k, the group effects for each latent
independent variable. We constrain (set equal) the value for one group to
be zero, so we can estimate the difference between that group and a second group, which we refer to as a group effect.
In the following LISREL matrix program we hypothesize that academic
and nonacademic boys are different in their reading and writing ability in
fifth and seventh grades. The first structured means program specifies the
number of groups (NG = 2), the first group’s (academic boys) sample size
(NO = 373), the number of observed variables (NI = 4), the type of matrix,
that is, a covariance matrix (MA = CM), and the first group’s covariance
matrix (CM) and means (ME). The second program only has to define the
second group’s (nonacademic boys) sample size (NO = 249), and the second group’s covariance matrix (CM) and means (ME). The means are what
defines a structured means program. Special features of this program are

Y102005.indb 405

3/22/10 3:27:15 PM

406

A Beginner’s Guide to Structural Equation Modeling

Writing7

err_w7

Reading7

err_r7

Writing5

err_w5

Reading5

err_r5

Verbal7

Verbal5

Figure 17.2
Structured means model.

setting TX = FR (t matrix of observed variable means) and KA = FI (k
matrix of latent variable means). This LISREL matrix program parallels the
LISREL–SIMPLIS program in chapter 13 for the structured means model,
that is, adding the LISREL OUTPUT command in the LISREL–SIMPLIS
program yields these same matrices and results (Figure 17.2).
The LISREL matrix structured means program is as follows:
Group: ACADEMIC
DA NI=4 NO=373 MA=CM NG=2
CM SY
281.349
184.219 182.821
216.739 171.699 283.289
198.376 153.201 208.837 246.069
ME
262.236 258.788 275.630 269.075
LA
R5 W5 R7 W7
MO NX=4 NK=2 TX=FR KA=FI
LK
V5 V7
FR LX(2,1) LX(4,2)
VA 1 LX(1,1) LX(3,2)
OU ND=2 AD=OFF
Group: NONACADEMIC
DA NI=4 NO=249 MA=CM
CM SY
174.485
134.468 161.869
129.840 118.836 228.449

Y102005.indb 406

3/22/10 3:27:15 PM

407

Matrix Approach to Structural Equation Modeling

102.194 97.767 136.058 180.460
ME
248.675 246.896 258.546 253.349
MO LX=IN TX=IN KA=FR TD=FR
LA
R5 W5 R7 W7
OU

The model parameters in the first group for t are set free (FR) and for k are
fixed (FI), so that the latent variable intercepts for the first group are fixed to
0. The estimate of the latent variable intercept in the second group (nonacademic boys) is therefore evaluated relative to 0 (academic boys intercept).
The structural model is represented as h2 = a2 + B2 h1 + z2 for both groups
separately for the null hypothesis H0: a academic boys = a nonacademic boys. The edited
and condensed structured means program output is as follows.
Group: ACADEMIC
LISREL Estimates (Maximum Likelihood)
LAMBDA-X EQUALS LAMBDA-X IN THE FOLLOWING GROUP
PHI

V5
V7

V5
-------220.06
(19.17)
11.48
212.11
(17.66)
12.01

V7
--------

233.59
(20.50)
11.40

THETA-DELTA
R5
-------50.15
(6.02)
8.34

W5
-------36.48
(4.28)
8.52

R7
-------51.72
(6.62)
7.82

W7
-------57.78
(6.05)
9.55

Squared Multiple Correlations for X - Variables
R5
-------0.81

W5
-------0.81

R7
-------0.82

W7
-------0.76

TAU-X EQUALS TAU-X IN THE FOLLOWING GROUP
Group Goodness-of-Fit Statistics
Contribution to Chi-Square = 4.15
Percentage Contribution to Chi-Square = 41.00

Y102005.indb 407

3/22/10 3:27:15 PM

408

A Beginner’s Guide to Structural Equation Modeling

Root Mean Square Residual (RMR) = 6.07
Standardized RMR = 0.025
Goodness-of-Fit Index (GFI) = 0.99
Group: NONACADEMIC
LISREL Estimates (Maximum Likelihood)
LAMBDA-X

R5
W5
R7
W7

KSI 1
-------1.00
0.84
(0.02)
34.35
- - -

KSI 2
-------- - 1.00
0.89
(0.03)
31.95

PHI

KSI 1

KSI 2

KSI 1
-------156.34
(16.19)
9.66
126.96
(14.22)
8.93

KSI 2
--------

153.73
(18.03)
8.53

THETA-DELTA
R5
-------23.25
(6.23)
3.73

W5
-------42.80
(5.64)
7.59

R7
-------65.67
(9.87)
6.65

W7
-------67.36
(8.74)
7.71

Squared Multiple Correlations for X-Variables
R5
-------0.87

W5
-------0.72

R7
-------0.70

W7
-------0.65

W5
-------258.67

R7
-------275.71

W7
-------268.98

TAU-X
R5
-------262.37

Y102005.indb 408

3/22/10 3:27:15 PM

409

Matrix Approach to Structural Equation Modeling

(0.84)
312.58

(0.70)
366.96

(0.87)
317.77

(0.80)
338.00

KAPPA
KSI 1
--------13.80
(1.18)
-11.71

KSI 2
--------17.31
(1.24)
-13.99

We obtain the latent variable mean differences from the kappa matrix,
where the nonacademic boys were below the academic boys in reading
and writing at both the fifth grade (KSI 1) and seventh grade (KSI 2). Our
model-fit indices indicate an acceptable theoretical model:
Global Goodness-of-Fit Statistics
Degrees of Freedom = 6
Minimum Fit Function Chi-Square = 10.11 (P = 0.12)
Normal Theory Weighted Least Squares Chi-Square = 9.96
(P = 0.13)
Estimated Noncentrality Parameter (NCP) = 3.96
90 Percent Confidence Interval for NCP = (0.0 ; 16.79)
Minimum Fit Function Value = 0.016
Population Discrepancy Function Value (F0) = 0.0064
90 Percent Confidence Interval for F0 = (0.0 ; 0.027)
Root Mean Square Error of Approximation (RMSEA) = 0.046
90 Percent Confidence Interval for RMSEA = (0.0 ; 0.095)
P-Value for Test of Close Fit (RMSEA < 0.05) = 0.27
Expected Cross-Validation Index (ECVI) = 0.087
90 Percent Confidence Interval for ECVI = (0.068 ; 0.095)
ECVI for Saturated Model = 0.032
ECVI for Independence Model = 3.15
Chi-Square for Independence Model with 12 Degrees of
Freedom = 1947.85
Independence AIC = 1963.85
Model AIC = 53.96
Saturated AIC = 40.00
Independence CAIC = 2007.31
Model CAIC = 173.48
Saturated CAIC = 148.66
Normed Fit Index (NFI) = 0.99
Nonnormed Fit Index (NNFI) = 1.00
Parsimony Normed Fit Index (PNFI) = 0.50
Comparative Fit Index (CFI) = 1.00
Incremental Fit Index (IFI) = 1.00
Relative Fit Index (RFI) = 0.99
Critical N (CN) = 1031.60

Y102005.indb 409

3/22/10 3:27:16 PM

410

A Beginner’s Guide to Structural Equation Modeling

Group Goodness-of-Fit Statistics
Contribution to Chi-Square = 5.97
Percentage Contribution to Chi-Square = 59.00
Root Mean Square Residual (RMR) = 7.69
Standardized RMR = 0.042
Goodness-of-Fit Index (GFI) = 0.99

17.4.4 Interaction Models
In chapter 16 we discussed four different types of interaction models: categorical, nonlinear, continuous observed variable, and latent variable (Schumacker
& Marcoulides, 1998). In this chapter we present the LISREL matrix program
using latent variables that parallels the interaction latent variable approach
in Chapter 16 using LISREL–SIMPLIS, except for slight differences in the
standard errors. The matrix approach to latent variable interactions requires
the understanding and use of nonlinear constraints, which has made it difficult for most SEM researchers (Jöreskog & Yang, 1996).
The latent variable interaction approach in LISREL matrix notation for
Figure 17.3 is h1 = g 1 x1 + g 2 x2 + g 3 x3 + z1, where h1 is the latent dependent
variable, x1 and x2 are the main-effect latent independent variables, x3 is
the interaction-effect formed by multiplying x1 and x2, g 1 and g 2 are the
structure coefficients for the main-effect latent independent variables, g 3 is

V4
V5

KS1
gamma1

V6
V47
V58

KSI1*KSI2

V69
V7

gamma3

gamma2

KSI2

V8
V9
Figure 17.3
Latent variable interaction (continuous variable approach).

Y102005.indb 410

3/22/10 3:27:16 PM

Matrix Approach to Structural Equation Modeling

411

the structure coefficient for the interaction-effect latent independent variable, and z1 is the error term in the structural equation. Notice that the
relationship between h1 and x3 is itself linear. The structure of the interaction model emerges as a logical extension of the measurement model for
x1 and x2. The basic measurement model is X = Λx + d, where X is a vector
of observed variables, Λ is a matrix of factor loadings, and d is a vector of
measurement error terms. The covariance matrices of these common and
unique factors are Φ and Θd, respectively.
Kenny and Judd (1984) used simple algebraic substitution to develop
their model of multiplicative interaction effects (Hayduk, 1987). Basically,
given two latent independent variables, the models are X1 = λ1 x1 + d 1 and
X2 = λ2 x2 + d 2. The interaction effect or product is X3 = X1 X2, indicated in
the model as X3 = λ1 λ2 x1 x2 + λ1 x1 d 2 + λ2 x2 d 1 + d 1 d 2, or X3 = λ3 x3 + λ1 x4 +
λ2 x5 + d 3, where x3 = x1 x2, x4 = x1 d 2, x5 = x2 d 1, d 3 = d 1 d 2, and λ3 = λ1 λ2. All
of these new latent variables are mutually uncorrelated and uncorrelated
with all other latent variables in the model.
In order to incorporate this interaction effect into the structural equation model, we need to specify X3 as a function of latent variables whose
variances and covariance terms reflect these relationships. This involves
specifying some model parameters as nonlinear functions of other
parameters. In the LISREL program, these types of nonlinear constraints
are indicated by using the VA (value), EQ (equality), and CO (constraint)
commands. For example, the Kenny–Judd interaction model implies that
s 2(x3) = s 2(x1) s 2(x2) + s( x1 x2)2. This relationship using the CO command
line is specified as CO PH(3,3) = PH(1,1) * PH(2,2) + PH(2,1) * * 2. Similarly,
their model implies that s 2(x4) = s 2(x1) s 2(d 2), and this relationship is specified as CO PH(4,4) = PH(1,1) * TD(2,2).
We demonstrate the Kenny and Judd (1984) approach by creating a
simulated data set of nine multivariate normal variables and three
product indicant variables for 500 participants using a PRELIS program
(mvdata1.pr2):
Generate multivariate normal variables – LISREL
DA NO=500
NE X1=NRAND; NE X2=NRAND; NE X3=NRAND
NE X4=NRAND; NE X5=NRAND; NE X6=NRAND
NE X7=NRAND; NE X8=NRAND; NE X9=NRAND
NE V1=X1
NE V2=.378*X1+.925*X2
NE V3=.320*X1+.603*X2+.890*X3
NE V4=.204*X1+.034*X2+.105*X3+.904*X4
NE V5=.076*X1+.113*X2+.203*X3+.890*X4+.925*X5
NE V6=.111*X1+.312*X2+.125*X3+.706*X4+.865*X5+.905*X6
NE V7=.310*X1+.124*X2+.310*X3+.222*X4+.126*X5+.555*X6+.897*X7
NE V8=.222*X1+.111*X2+.412*X3+.312*X4+.212*X5+.312*X6+.789*X7+.899*X8

Y102005.indb 411

3/22/10 3:27:16 PM

412

A Beginner’s Guide to Structural Equation Modeling

NE V9=.321*X1+.214*X2+.124*X3+.122*X4+.234*X5+.212*X6+.690*X7+.789*X8+.907*X9
NE V47=V4*V7
NE V58=V5*V8
NE V69=V6*V9
CO ALL
SD X1-X9
OU MA=CM CM=INTERACT.CM ME=INTERACT.ME RA=INTERACT.PSF XM
IX=784123

Although the nine observed variables were created as multivariate normal data, the product indicant variables are typically not multivariate normal. The summary statistics do indicate that the nine observed variables
are univariate normal, but that the three product indicant variables have
skewness and kurtosis, that is, are nonnormal (boldfaced). In LISREL,
maximum likelihood estimation (ML) is the default, and it appears to
work well under mild violations of multivariate normality in the interaction latent variable model. We used the same random number seed as
before so the data could be reproduced (IX = 784123).
PRELIS Computer Output
Univariate Summary Statistics for Continuous Variables
Variable
V1
V2
V3
V4
V5
V6
V7
V8
V9
V47
V58
V69

Mean
-0.061
0.007
-0.018
-0.015
-0.013
0.011
-0.065
-0.041
0.005
0.325
0.670
0.584

St. Dev.
0.976
1.071
1.105
0.956
1.351
1.543
1.192
1.491
1.595
1.143
2.179
2.754

T-Value
-1.394
0.142
-0.368
-0.359
-0.209
0.163
-1.222
-0.615
0.075
6.356
6.877
4.745

Skewness
0.191
-0.047
0.175
-0.200
-0.003
0.171
-0.081
0.127
0.058
0.958
1.916
2.304

Kurtosis
0.048
0.280
0.441
-0.158
0.168
0.528
-0.350
0.092
0.514
3.861
8.938
15.266

Test of Univariate Normality for Continuous Variables
Skewness and
Kurtosis
ChiZ-Score P-Value Z-Score P-Value Square P-Value
1.749
0.080
0.321
0.748
3.163
0.206
-0.432
0.666
1.256
0.209
1.764
0.414
1.608
0.108
1.811
0.070
5.866
0.053

Skewness
Variable
V1
V2
V3

Y102005.indb 412

Kurtosis

3/22/10 3:27:17 PM

413

Matrix Approach to Structural Equation Modeling

V4
V5
V6
V7
V8
V9
V47
V58
V69

-1.833
-0.031
1.571
-0.746
1.165
0.531
7.573
12.103
13.428

0.067
0.975
0.116
0.456
0.244
0.595
0.000
0.000
0.000

-0.695
0.829
2.082
-1.865
0.513
2.039
7.085
9.622
11.101

0.487
0.407
0.037
0.062
0.608
0.041
0.000
0.000
0.000

3.844
0.688
6.802
4.034
1.620
4.438
107.539
239.070
303.539

0.146
0.709
0.033
0.133
0.445
0.109
0.000
0.000
0.000

The PRELIS program saves three files, a covariance matrix (interact.cm),
means (interact.me), and a PRELIS system file (interact.psf). The LISREL
program inputs the files with the covariance matrix and means.
The LISREL program to run the data for the model in Figure 17.3 is:
Fitting Model to Mean Vector and Covariance Matrix
DA NI=12 NO=500
!The three interaction variables are added prior to
program analysis
LA
V1 V2 V3 V4 V5 V6 V7 V8 V9 V47 V58 V69
CM=interact.CM
ME=interact.ME
MO NY=3 NX=9 NE=1 NK=3 TD=SY TY=FR TX=FR KA=FR
FR LY(2) LY(3) GA(1) GA(2) GA(3) LX(2,1) LX(3,1) LX(5,2)
LX(6,2) PH(1,1)-PH(2,2)
FI PH(3,1) PH(3,2)
VA 1 LY(1) LX(1,1) LX(4,2) LX(7,3) !Should be same as
C SIMPLIS program for comparison
FI KA(1) KA(2)
CO LX(7,1)=TX(4)
CO LX(7,2)=TX(1)
CO LX(8,1)=TX(5)*LX(2,1)
CO LX(8,2)=TX(2)*LX(5,2)
CO LX(8,3)=LX(2,1)*LX(5,2)
CO LX(9,1)=TX(6)*LX(3,1)
CO LX(9,2)=TX(3)*LX(6,2)
CO LX(9,3)=LX(3,1)*LX(6,2)
CO PH(3,3)=PH(1,1)*PH(2,2)+PH(2,1)**2
CO TD(7,1)=TX(4)*TD(1,1)
CO TD(7,4)=TX(1)*TD(4,4)
CO TD(7,7)=TX(1)**2*TD(4,4)+TX(4)**2*TD(1,1)+PH(1,1)*TD(4,4)+
C PH(2,2)*TD(1,1)+TD(1,1)*TD(4,4)
CO TD(8,2)=TX(5)*TD(2,2)
CO TD(8,5)=TX(2)*TD(5,5)
CO TD(8,8)=TX(2)**2*TD(5,5)+TX(5)**2*TD(2,2)+LX(2,1)**2*PH(1,
1)*TD(5,5)+
C LX(5,2)**2*PH(2,2)*TD(2,2)+TD(2,2)*TD(5,5)

Y102005.indb 413

3/22/10 3:27:17 PM

414

A Beginner’s Guide to Structural Equation Modeling

CO TD(9,3)=TX(6)*TD(3,3)
CO TD(9,6)=TX(3)*TD(6,6)
CO TD(9,9)=TX(3)**2*TD(4,4)+TX(6)**2*TD(3,3)+LX(3,1)**2*PH(1,
1)*TD(6,6)+
C LX(6,2)**2*PH(2,2)*TD(3,3)+TD(3,3)*TD(6,6)
CO KA(3)=PH(2,1)
CO TX(7)=TX(1)*TX(4)
CO TX(8)=TX(2)*TX(5)
CO TX(9)=TX(3)*TX(6)
OU AD=OFF IT=500 EP=0.001 IM=3 ND=3

The CO command (placing proper constraints in the model) is what
becomes difficult to navigate in creating the matrix programs for latent
variable interaction models. Discussions of different latent variable
interaction models and related issues can be found in Marcoulides and
Schumacker (1996, 2001) and Schumacker and Marcoulides (1998).
Given the LISREL matrix program with a latent variable interaction
term, several matrices need to be specified. The structural equation with
the two main-effect latent variables and the interaction-effect latent variable is as follows:

η = α + γ 1ξ1 + γ 2ξ2 + γ 3ξ1ξ2 + ζ
The measurement model with Y observed variables is defined as follows:
Y = Ty + Λ yη + ε
The matrices for the Y observed variable measurement model are specified as:
(y)
 y1   τ 1   1 
 ε1 
 y  =  τ (y)  +  λ (y)  η +  ε  ,
 2  2   2 
 2
 y   τ ( y )   λ ( y ) 
 ε 
3
3
 3   3 

where the theta–epsilon error matrix specified as follows:
Θ ε = diag(θ1ε , θ 2ε , θ 3ε )
The measurement model for the X observed variables, which includes
both main-effects and the interaction-effect, is defined as follows:
X = Tx + Λ xξ + δ

Y102005.indb 414

3/22/10 3:27:18 PM

415

Matrix Approach to Structural Equation Modeling

The matrices for the X observed variable measurement model are specified as follows:
 x1   τ 1 


 
 x2   τ 2   1

 
 
 x3   τ 3   λ 2

  λ
 
 x  τ   3
 4   4   0

 
 
 x5  =  τ 5  +  0

  0
 
 x6   τ 6  

  τ4
 
 x1x4   τ 1τ 4   τ λ

  5 2
 
 x2 x5   τ 2τ 5   τ 3 λ3


 
 x x   τ τ 
3 6
3 6

0
0
0
1
λ5
λ6
τ1
τ 2 λ5
τ 3 λ6

 δ1 
 
δ2 
0 
 

0
δ3 

 
0 
  ξ1   δ 4 
0 

  
0   ξ2  +  δ 5 

  
0 
ξ1ξ2   δ 6 

1 
 

 δ7 
λ2 λ5 
 

λ 3 λ6 
δ8 
 
 δ 
9

with errors in the theta delta matrix Θd denoted as follows:
 θ1
 0

 0

 0
 0

 0
τ θ
 4 1
 0

 0

θ2
0
0
0
0
0
τ 5θ 2
0

θ3
0
0
0
0
0
τ 6θ 3

θ4
0
0
τ 1θ 4
0
0

θ5
0
0
τ 2θ 5
0

θ6
0
0
τ 3θ6

θ7
0
0

θ8
0







.






θ9 

The theta delta values for the observed interaction variables are calculated
as follows:

θ7 = τ 42θ1 + τ 12θ 4 + φ11θ 4 + φ22θ1 + θ1θ 4
θ 8 = τ 52θ 2 + τ 22θ 5 + λ22φ11θ 5 + λ52φ22θ 2 + θ 2θ 5
θ 9 = τ 62θ 3 + τ 32θ6 + λ32φ11θ6 + λ62φ22θ 3 + θ 3θ6

Y102005.indb 415

3/22/10 3:27:19 PM

416

A Beginner’s Guide to Structural Equation Modeling

The mean vector implied by the interaction of the exogenous latent variables is defined in the following kappa mean vector matrix:
0 
 
κ = 0 
 
 φ 
21
with the variance–covariance of the latent independent variables (ksi1
and ksi2) defined as follows:
 φ11
Φ =  φ21

 0

φ22
0




2 
φ11φ22 + φ21


We can now look for these matrices and their associated values in the
LISREL computer output.
LISREL Interaction Computer Output
The gamma matrix contains the three structure coefficients of interest for
the two main-effect latent variables [g 1 = .077 (.030), t = 2.60 and g 2 = .155
(.029), t = 5.378] and the interaction latent variable [g 3 = −.029 (.029), t =
–1.004]. The gamma coefficient for the latent variable interaction effect is
nonsignificant (t = −1.004). We should modify our theoretical model and
test main effects only. The edited and condensed LISREL computer output
is as follows:
LISREL Estimates (Maximum Likelihood)
LAMBDA-Y

V1
V2

Y102005.indb 416

ETA 1
-------1.000
2.080
(0.257)
8.097
2.532
(0.325)
7.788

3/22/10 3:27:19 PM

417

Matrix Approach to Structural Equation Modeling

LAMBDA-X
KSI 1
-------1.000
1.981
(0.091)
21.732
1.925
(0.090)
21.368
- - -

V4
V5
V6
V7
V8
V9

- (0.069)
21.741
-0.070
(0.035)
-2.027
-0.094
(0.083)
-1.142
0.020
(0.090)
0.226

V47
V58
V69

KSI 2
-------- - -

KSI 3
-------- - -

- -

1.000
1.658
(0.072)
22.874
1.493

- - -

0.013
(0.028)
0.471
-0.010
(0.062)
-0.153
-0.049
(0.067)
-0.729

1.000
3.285
(0.173)
18.960
2.875
(0.152)
18.940

KSI 2
-------0.155
(0.029)
5.378

KSI 3
--------0.029
(0.029)
-1.004

- -

GAMMA

ETA 1

KSI 1
-------0.077
(0.030)
2.602

Covariance Matrix of ETA and KSI

ETA
KSI
KSI
KSI

1
1
2
3

ETA 1
-------0.150
0.068
0.137
-0.012

KSI 1
--------

KSI 2
--------

KSI 3
--------

0.463
0.211
- -

0.784
- -

0.408

Mean Vector of Eta-Variables
ETA 1
--------0.006

Y102005.indb 417

3/22/10 3:27:20 PM

418

A Beginner’s Guide to Structural Equation Modeling

PHI
KSI 1
-------0.463
(0.043)
10.729
0.211
(0.021)
10.236
- -

KSI 1
KSI 2
KSI 3

KSI 2
--------

KSI 3
--------

0.784
(0.069)
11.283
- -

0.408
(0.041)
9.953

PSI
ETA 1
-------0.123
(0.029)
4.193
Squared Multiple Correlations for Structural Equations
ETA 1
-------0.179
THETA-EPS
V1
-------0.804
(0.053)
15.166

V2
-------0.502
(0.058)
8.618

V3
-------0.267
(0.074)
3.617

Squared Multiple Correlations for Y - Variables
V1
-------0.157

V2
-------0.563

V3
-------0.782

THETA-DELTA

V4
V5
V6

Y102005.indb 418

V4
V5
V6
V7
V8
V9
-------- -------- -------- -------- -------- -------0.458
(0.029)
16.044
- 0.045
(0.049)
0.931
- - 0.796

3/22/10 3:27:20 PM

419

Matrix Approach to Structural Equation Modeling

- -

(0.065)
12.238
- -

- -

0.647
(0.043)
15.186
- -

- -

V47

- -

V58

-0.032
(0.016)
-2.011
- -

- -

V69

- -

-0.002
(0.003)
-0.722
- -

0.009
(0.018)
0.471
- -

0.008
(0.037)
0.226

- -

0.105
(0.057)
1.840
- -

0.936
(0.070)
13.342
- -

- -0.001
(0.004)
-0.152
- -

- -0.031
(0.042)
-0.728

THETA–DELTA

V58

V47
-------0.957
(0.047)
20.519
- -

V69

- -

V47

V58
--------

0.293
(0.138)
2.129
- -

V69
--------

3.745
(0.200)
18.698

Squared Multiple Correlations for X - Variables
V4
-------0.503

V5
-------0.976

V6
-------0.683

V7
-------0.548

V8
-------0.954

V9
-------0.651

Squared Multiple Correlations for X - Variables
V47
-------0.300

Y102005.indb 419

V58
-------0.938

V69
-------0.474

3/22/10 3:27:20 PM

420

A Beginner’s Guide to Structural Equation Modeling

TAU-Y
V1
--------0.055
(0.044)
-1.263

V2
-------0.019
(0.048)
0.386

V3
--------0.004
(0.050)
-0.079

TAU-X
V4
-------0.013
(0.028)
0.471

V5
--------0.006
(0.038)
-0.153

V6
--------0.033
(0.045)
-0.729

V7
--------0.070
(0.035)
-2.027

V8
--------0.048
(0.042)
-1.144

V9
-------0.011
(0.047)
0.226

TAU-X
V47
--------0.001
(0.002)
-0.456

V58
-------0.000
(0.002)
0.151

V69
-------0.000
(0.002)
-0.215

KAPPA
KSI1
-------- -

KSI2
-------- -

KSI3
-------0.211
(0.021)
10.236

Goodness-of-Fit Statistics
Degrees of Freedom = 59
Minimum Fit Function Chi-Square = 403.462 (P = 0.0)
Normal Theory Weighted Least Squares Chi-Square = 365.186
(P = 0.0)
Estimated Non-centrality Parameter (NCP) = 306.186
90 Percent Confidence Interval for NCP = (249.618 ; 370.256)
Minimum Fit Function Value = 0.809
Population Discrepancy Function Value (F0) = 0.614
90 Percent Confidence Interval for F0 = (0.500 ; 0.742)
Root Mean Square Error of Approximation (RMSEA) = 0.102
90 Percent Confidence Interval for RMSEA = (0.0921 ; 0.112)

Y102005.indb 420

3/22/10 3:27:20 PM

Matrix Approach to Structural Equation Modeling

421

P-Value for Test of Close Fit (RMSEA < 0.05) = 0.000
Expected Cross-Validation Index (ECVI) = 0.944
90 Percent Confidence Interval for ECVI = (0.763 ; 1.005)
ECVI for Saturated Model = 0.313
ECVI for Independence Model = 6.300
Chi-Square for Independence Model with 66 Degrees of
Freedom = 3119.580
Independence AIC = 3143.580
Model AIC = 471.186
Saturated AIC = 156.000
Independence CAIC = 3206.156
Model CAIC = 747.560
Saturated CAIC = 562.739
Normed Fit Index (NFI) = 0.871
Nonnormed Fit Index (NNFI) = 0.874
Parsimony Normed Fit Index (PNFI) = 0.778
Comparative Fit Index (CFI) = 0.887
Incremental Fit Index (IFI) = 0.887
Relative Fit Index (RFI) = 0.855
Critical N (CN) = 108.807
Root Mean Square Residual (RMR) = 0.142
Standardized RMR = 0.0636
Goodness-of-Fit Index (GFI) = 0.893
Adjusted Goodness-of-Fit Index (AGFI) = 0.859
Parsimony Goodness-of-Fit Index (PGFI) = 0.676

17.5 Summary
This chapter presented the eight basic matrices used in structural equation modeling, plus two new matrices, tau and kappa. We also discussed
that for any structural equation model, parameters in these matrices must
be free, fixed, or constrained for model identification, model estimation,
and model testing. We presented the matrix notation by providing five
different SEM models: our theoretical model in chapter 10, a path model,
a multiple-sample model, a structured means model, and an interaction
model. We presented these same models in earlier chapters using LISREL–
SIMPLIS, and displayed them in matrix form in this chapter for comparative purposes. If you wish, simply add the LISREL OUTPUT command

Y102005.indb 421

3/22/10 3:27:20 PM

422

A Beginner’s Guide to Structural Equation Modeling

to these programs to output the matrices presented in this chapter. We
firmly believe that once you master the matrix notation, you will better
understand structural equation modeling.
Exercise
1. The National Science Foundation (NSF) is encouraging students
to seek academic degrees and careers in science, mathematics,
and engineering in the United States. Research has shown a gender difference in science, mathematics, and engineering participation. A key area of study is to investigate what factors influence
these gender differences. A latent variable model is hypothesized
to investigate factors that influence gender differences because
previous research indicated variables such as characteristics of
students in science, mathematics, and engineering.
A structural equation model with two exogenous latent variables measured by six observed variables is hypothesized to
predict two endogeneous latent variables measured by five
observed variables. The first independent latent variable, x1 =
Family Background, is measured by three variables: X1 = family income, X2 = father’s education, and X3 = mother’s education.
The other independent latent variable, x2 = Encouragement, is
measured by three variables: X4 = personal encouragement,
X5 = institutional characteristics, and X6 = admission status.
Students’ characteristics, h 1 = Students’ Characteristics, is
measured by three variables: Y1 = cognitive abilities, Y2 = interpersonal skills, and Y3 = motivation. The other endogenous
variable, h2 = Aspirations, is measured by two variables: Y4 =
occupational aspiration and Y5 = educational aspiration.
The hypothesized structural equation model represents a
two-step approach: measurement (confirmatory factor analysis)
and structural model. The structural model depicts the relationships between four latent variables: x1 = Family Background,
x2 = Encouragement, h1 = Students’ Characteristics, and h2 =
Aspirations. The structural model is

Students’ Characteristics = Family Background + Encouragement
+ Aspirations + error
Aspirations = Family Background + Encouragement + error.
With this information, you should be able to do the following:
1.
2.

Y102005.indb 422

Diagram the structural equation model.
Write the measurement equations using the variable
names.

3/22/10 3:27:20 PM

Matrix Approach to Structural Equation Modeling

3.
4.
5.
6.
7.

423

Write the measurement equations using LISREL matrix
notation.
Write the structural equations using the variable names.
Write the structural equations using LISREL matrix
notation.
Create the matrices for the measurement model.
Create the matrices for the structural model.

References
Hayduk, L. A. (1987). Structural equation modeling with LISREL: Essentials and
advances. Baltimore, MD: Johns Hopkins University Press.
Jöreskog, K. G., & Sörbom, D. (1996). LISREL8 user’s reference guide. Chicago, IL:
Scientific Software International.
Jöreskog, K. G., & Yang, F. (1996). Non-linear structural equation models: The
Kenny-Judd model with interaction effects. In G. A. Marcoulides & R. E.
Schumacker (Eds.), Advanced structural equation modeling: Issues and techniques
(pp. 57–88). Mahwah, NJ: Lawrence Erlbaum.
Kenny, D. A., & Judd, C. M. (1984). Estimating the non‑linear and interactive effects
of latent variables. Psychological Bulletin, 96, 201–210.
Marcoulides, G., & Schumacker, R. E. (Eds.). (1996). Advanced structural equation
modeling: Issues and techniques. Mahwah, NJ: Lawrence Erlbaum.
Marcoulides, G., & Schumacker, R. E. (Eds.). (2001). New developments and techniques in structural equation modeling. Mahwah, NJ: Lawrence Erlbaum.
Schumacker, R. E., & Marcoulides, G. A. (1998). Interaction and nonlinear effects in
structural equation modeling. Mahwah, NJ: Lawrence Erlbaum.

Y102005.indb 423

3/22/10 3:27:20 PM

Appendix A: Introduction
to Matrix Operations
Structural Equation Modeling performs calculations using several different matrices. The matrix operations to perform the calculations involve
addition, subtraction, multiplication, and division of elements in the different matrices.* We present these basic matrix operations, followed by a
simple multiple regression example.

Matrix Definition
A matrix is indicated by capital letters (e.g., A, B, or R) and takes the form:
3
A22 = 
5

5

6

The matrix can be rectangular or square-shaped, and contains an array of
numbers. A correlation matrix would be a square matrix with the value
of 1.0 in the diagonal and variable correlations in the off-diagonal. A correlation matrix is symmetrical because the correlation coefficients in the
lower half of the matrix are the same as the correlation coefficients in the
upper half of the matrix. [Note: we usually only report the diagonal values
and the correlations in the lower half of the matrix.] For example:
1.0

R33 = .30
.50


.30
1.0
.60

.50 

.60  ,
1.0 

but we report the following as a correlation matrix:
1.0
.30 1.0
.50 .60

1.0

Walter L. Sullins (1973). Matrix algebra for statistical applications, Danville, IL: The Interstate
Printers & Publishers, Inc.

425

Y102005.indb 425

3/22/10 3:27:21 PM

426

Appendix A: Introduction to Matrix Operations

Matrices have a certain number of rows and columns. The A matrix
above has two rows and two columns. The order of a matrix is the size of
the matrix, or number of rows times the number of columns. The order
of the A matrix is 22, and shown as subscripts, where the first subscript
is the number of rows, and second subscript is the number of columns.
When we refer to elements in the matrix, we use row and column designations to identify the location of the element in the matrix. The location
of an element has a subscript using the row number first, followed by the
column number. For example, the correlation r = .30 is in the R 21 matrix
location or row 2, column 1.

Matrix Addition and Substraction
Matrix addition adds corresponding elements in two matrices, while
matrix subtraction subtracts corresponding elements in two matrices.
Consequently, the two matrices must have the same order (number of
rows and columns), so we can add A32 + B32 or subtract A32 – B32. In the following example, Matrix A elements are added to Matrix B elements:
3

1
9


5
6
1

2 1
 
0  + 2
2   0

−3
1
7

5  4
 
3  = 3
−3   9

2
7
8

7

3
−1

Matrix Multiplication
Matrix multiplication is not as straight forward as matrix addition and
subtraction. For a product of matrices we indicate A • B or AB. If A is an
m × n matrix and B is an n × p matrix, then AB is a m × p matrix of rows
and columns. The number of columns in the first matrix must match the
number of rows in the second matrix to be compatible and permit multiplication of the elements of the matrices. The following example will illustrate how the row elements in the first matrix (A) are multiplied times
the column elements in the second matrix (B) to yield the elements in the
third matrix C.
c11 = 1•2 + 2•1 = 2 + 2 = 4
c12 = 1•4 + 2•8 = 4 + 16 = 20
c13 = 1•6 + 2•7 = 6 + 14 = 20

Y102005.indb 426

3/22/10 3:27:22 PM

427

Appendix A: Introduction to Matrix Operations

c21 = 3•2 + 5•1 = 6 + 5 = 11
c22 = 3•4 + 5•8 = 12 + 40 = 52
c23 = 3•6 + 5•7 = 18 + 35 = 53
1
A•B = 
3

2  2
•
5 1

6  4
=
7   11

4
8

20
52

20 

53 

Matrix C is:
4
C=
11

20
52

20 

53 

It is important to note that matrix multiplication is noncommutative (i.e.,
AB ≠ BA.) The order of operation in multiplying elements of the matrices is therefore very important. Matrix multiplication, however, is associate [i.e., A (BC) = (AB) C] because the order of matrix multiplication is
maintained.
A special matrix multiplication is possible when a single number is multiplied times the elements in a matrix. The single number is called a scalar.
The scalar is simply multiplied times each of the elements in the matrix.
For example,
3
D = 2
4

4  6
=
6  8

8

12 

Matrix Division
Matrix division is similar to matrix multiplication with a little twist. In
regular division, we divide the numerator by the denominator. However,
we can also multiply the numerator by the inverse of the denominator. For
example, in regular division, 4 is divided by 2; however, we get the same
results if we multiply 4 by ½. Therefore, matrix division is simply A/B or
A•1/B = AB−1. The special designation of the B−1 matrix is called the inverse
of the B matrix.
Matrix division requires finding the inverse of a matrix, which involves
computing the determinant of a matrix, the matrix of minors, and the matrix
of cofactors. We then create a transposed matrix and an inverse matrix, which
when multiplied yield an identity matrix. We now turn our attention to
finding these values and matrices involved in matrix division.

Y102005.indb 427

3/22/10 3:27:23 PM

428

Appendix A: Introduction to Matrix Operations

Determinant of a Matrix
The determinant of a matrix is a unique number (not a matrix) that uses
all the elements in the matrix for its calculation, and is a generalized variance for that matrix. For our illustration we will compute the determinant
of a 2 by 2 matrix; leaving higher order matrix determinant computations
for high-speed computers. The determinant is computed by cross multiplying the elements of the matrix:
a
A=
c

b

d

so, the determinant of A = ad − cb.
For example,
2
A=
3

5

6

so, the determinant of A = 2•6 – 3•5 = −3.
Matrix of Minors
Each element in a matrix has a minor. To find the minor of each element, simply
draw a vertical and a horizontal line through that element to form a matrix
with one less row and column. We next calculate the determinants of these
minor matrices, and then place them in a matrix of minors. The matrix of minors
would have the same number of rows and columns as the original matrix.
The matrix of minors for the following 3 by 3 matrix would be computed
as follows:

Y102005.indb 428

1

A =  −2
3


6
7
−1

−3 

1
4 

7
M11 
 −1

1
 = (7)(4) − (1)( − l) = 29
4 

 −2
M12 
 3

1
 = ( − 2)(4) − (1)(3) = −11
4 

 −2
M13 
 3


 = ( − 2)( − l) − (7)(3) = −19
−1

3/22/10 3:27:24 PM

Appendix A: Introduction to Matrix Operations

6
M 21 
 −1

− 3
 = (6) (4) − ( − 1)( − 3) = 21
4 

1
M 22 
 3

− 3
 = (1)(4) − ( − 3)(3) = 13
4 

1
M 23 
 3

6 
 = (1)( − l) − (6)(3) = −19
− 1

6
M 31 
7

− 3
 = (6)(1) − ( − 3)(7) = 27
1 

1
M 32 
 −2

−3 
 = (1)(1) − ( − 3)( − 2) = −5
1

1
M 33 
 −2
AMinors

429

6
 = (1)(7) − (6)( − 2) = 19
7 

 29

=  21
 27


−11
13
−5

−19 

−19 
19 

Matrix of Cofactors
A matrix of cofactors is created by multiplying the elements of the matrix of
minors by (−l) for i + j elements, where i = row number of the element and
j = column number of the element. Place these values in a new matrix,
called a matrix of cofactors.
An easy way to remember this multiplication rule is to observe the
matrix below. Start with the first row and multiply the first entry by (+),
second entry by (−), third by (+), and so on to the end of the row. For the
second row start multiplying by (−), then (+), then (−), and so on. All odd
rows begin with + sign and all even rows begin with − sign.
+
–
+
–

Y102005.indb 429

–
+
–
+

+
–
+
–

3/22/10 3:27:25 PM

430

Appendix A: Introduction to Matrix Operations

We now proceed by multiplying elements in the matrix of minors by −1 for
the i + j elements.

AMinors

 +1

=  −1
 +1


−1
+1
−1

+1  29

−1  21
+1  27

−11
13
−5

−199 

−19 
19 

to obtain the matrix of cofactors:

CCofactors

 29

=  −21
 27


11
13
5

−19 

19 
19 

Determinant of Matrix Revisited
The matrix of cofactors makes finding the determinant of any size matrix
easy. We multiply elements in any row or column of our original A matrix,
by any one corresponding row or column in the matrix of cofactors to compute the determinant of the matrix. We can compute the determinant
using any row or column, so rows with zeroes makes the calculation of
the determinant easier. The determinant of our original 3 by 3 matrix (A)
using the 3 by 3 matrix of cofactors would be:
det A = a11c11 + a12 c12 + a13c13
Recall that matrix A was:
1

A =  −2
3


6
7
−1

−3 

1
4 

The matrix of cofactors was:

CCofactors

 29

=  −21
 27


11
13
5

−19 

19 
19 

So, the determinant of matrix A, using the first row of both matrices is.
det A = (1)(29) + (6)(11) + (−3)(−19) = 152

Y102005.indb 430

3/22/10 3:27:27 PM

431

Appendix A: Introduction to Matrix Operations

We also could have used the second columns of both matrices and obtained
the same determinant value:
det A = (6)(11) + (7 )(13) + (−1)(5) = 152
Two special matrices, we have already mentioned, also have determinants: diagonal matrix and triangular matrix. A diagonal matrix is a
matrix which contains zero or nonzero elements on its main diagonal,
but zeroes everywhere else. A triangular matrix has zeros only either
above or below the main diagonal. To calculate the determinants of
these matrices, we only need to multiply the elements on the main
diagonal. For example, the following triangular matrix K has a determinant of 96.
2

4
K=
 −1

 3

0
1
5
9

0
0
6
−2

0

0
0

8 

This is computed by multiplying the diagonal values in the matrix:
det K = (2)(1)(6)(8) = 96.

Transpose of a Matrix
The transpose of a matrix is created by taking the rows of an original
matrix C and placing them into corresponding columns of a transpose
matrix, C’. For example:
 29

C =  −21
 27


11
13
5

 29

C ′ =  11
 −19


−21
13
19

−19 

19 
19 
27 

5
19 

The transposed matrix of the matrix of cofactors is now given the special
term adjoint matrix, designated as Adj(A). The adjoint matrix is important
because we use it to create the inverse of a matrix, our final step in matrix
division operations.

Y102005.indb 431

3/22/10 3:27:28 PM

432

Appendix A: Introduction to Matrix Operations

Inverse of a Matrix
The general formula for finding an inverse of a matrix is one over the
determinant of the matrix times the adjoint of the matrix:
A −1 = [1 /det A] ADJ ( A)
Since we have already found the determinant and adjoint of A, we find the
inverse of A as follows:
 29


1

A −1 = 
11
 152  
 −19


−21
13
19

27   .191
 
5  =  .072
19   −.125

−.138
.086
.125

.178 

.033 
.125 

An important property of the inverse of a matrix is that if we multiply
its elements by the elements in our original matrix, we should obtain an
identity matrix. An identity matrix will have 1.0 in the diagonal and zeroes
in the off-diagonal. The identity matrix is computed as:
A A −1 = I
Because we have the original matrix of A and the inverse of matrix A, we
multiply elements of the matrices to obtain the identity matrix, I:
1

AA =  −2
3

−1

6
7
−1

−3   .191
 
1  ∗  .072
4   −.125

−.138
.086
.125

.178   1
 
.033  =  0
.125   0

0
1
0

0

0
1 

Matrix Operations in Statistics
We now turn our attention to how the matrix operations are used to compute statistics. We will only cover the calculation of the Pearson correlation and provide the matrix approach in multiple regression, leaving more
complicated analyses to computer software programs.
Pearson Correlation (Variance–Covariance Matrix)
In the book, we illustrated how to compute the Pearson correlation coefficient from a variance–covariance matrix. Here, we demonstrate the matrix

Y102005.indb 432

3/22/10 3:27:29 PM

433

Appendix A: Introduction to Matrix Operations

approach. An important matrix in computing correlations is the sums of
squares and cross-products matrix (SSCP). We will use the following pairs
of scores to create the SSCP matrix.
X1

5
4
6

1
3
5

The mean of X1 is 5 and the mean of X2 is 3. We use these mean values
to compute deviation scores from each mean. We first create a matrix of
deviation scores, D:
5

D = 4
6


1 5
 
3 − 5
5   5

3  0
 
3  =  −1
3   1

−2 

0
2 

Next, we create the transpose of matrix D, D’:
0
D′ = 
 −2

−1
0

1

2

Finally, we multiply the transpose of matrix D times the matrix of deviation scores to compute the sums of squares and cross-products matrix:
SSCP = D’ * D
0
SSCP = 
 −2

−1
0

0
1 
 ∗ −1
2 
1


−2 
 2
0 =
2
2  

2

8

The sums of squares are along the diagonal of the matrix, and the sum
of squares cross-products are on the off-diagonal. The matrix multiplications are provided below for the interested reader.
(0)(0) + (−1)(−1) + (1)(1) = 2 [sums of squares = (0² + −1² + 1²)]
(−2)(0) + (0)(−1) + (2)(1) = 2 [sum of squares cross product]
(0)(−2) + (−1)(0) + (1)(2) = 2 [sum of squares cross product]
(−2)(−2) + (0)(0) + (2)(2) = 8[sums of squares = (−2² + 0² + 2²)]

Y102005.indb 433

3/22/10 3:27:29 PM

434

Appendix A: Introduction to Matrix Operations

2
SSCP = 
2

2

8

Sum of squares in diagonal of matrix

Variance–Covariance Matrix
Structural equation modeling uses a sample variance–covariance matrix
in its calculations. The SSCP matrix is used to create the variance–covariance matrix, S:
S=

SSCP
n−1

In matrix notation this becomes ½ times the matrix elements:
Covariance terms in the off-diagonal
1  2 2  1 1 
of matrix
S = ∗
=

2  2 8  1 4 
Variance of variables in diagonal
of matrix
We can now calculate the Pearson correlation coefficient using the
basic formula of covariance divided by the square root of the product
of the variances.
r=

CovarianceX 1X 2
VarianceX 1 ∗ VarianceX 2

1
1∗ 4

1
= .50
2

Multiple Regression
The multiple linear regression equation with two predictor variables is:
y = β 0 + β1 x 1 + β 2 x 2 + e i
where y is the dependent variable, x1 and x2 the two predictor variables,
and

β0 is the regression constant or y-intercep
pt,
β1 and β2 are the regression weights to be estimated,
and e is the error of preediction.

Y102005.indb 434

3/22/10 3:27:31 PM

435

Appendix A: Introduction to Matrix Operations

Given the data below, we can use matrix algebra to estimate the regression weights:
y

3
2
4
5
8

2
3
5
7
8

1
5
3
6
7

We model each subject’s y score as a linear function of the betas:
y1 = 3 = 1β0 + 2 β1 + 1 β2 + e1
y 2 = 2 = 1β0 + 3 β1 + 5 β2 + e2
y 3 = 4 = 1β0 + 5 β1 + 3 β2 + e3
y 4 = 5 = 1β0 + 7 β1 + 6 β2 + e4
y 5 = 8 = 1β0 + 8 β1 + 7 β2 + e5
This series of equations can be expressed as a single matrix equation:
y=

 3  1
  
 2  1
  
y =  4  = 1
  
 5  1
  
 8  1
  

2
3
5
7
8

β + e
1

5

3

6

7 

 e1 
 
 β0   e2 
   
 β1  +  e 3 
   
β  e 
 2  4
 
 e5 

The first column of matrix X are 1s, which compute the regression constant.
In matrix form, the multiple linear regression equation is y = Xβ + e.
Using calculus, we translate this matrix to solve for the regression weights:

βˆ = (X ' X )−1 X ' y

Y102005.indb 435

3/22/10 3:27:32 PM

436

Appendix A: Introduction to Matrix Operations

The matrix equation is:
X′



  1 1 1 1 1



βˆ =   2 3 5 7 8 


 1 5 3 6 7 



1

1

1

1

1


X′
2
3
5
7
8

1 

5 


3 

6 

7  

−1

 1 1 1 1 1


* 2 3 5 7 8 


1 5 3 6 7 



y
3 
 
2 
 
4
 
5 
 
8 
 

We first compute X’ X and then compute X’y
 5 25 22 
 22 




X ′X =  25 151 130  and X ′ y = 131




 22 130 120 
111




Next, we create the inverse of X’X, where 1016 is the determinant of X’X.

(X 'X )−1 =

1220 −140 −72 


1
 −140 116 −100 
1016 

 −72 −100 130 



Finally, we solve for the X1 and X2 regression weights:

β̂ =

1220

1
 −140
1016 
 −72


− 140
116
− 100

− 72 

− 100 

130 

 22 
.50 




131 = 1 




111
 −.25 





The multiple regression equation is:
ŷ i = .50 + 1X1 − .25 X 2
We use the multiple regression equation to compute predicted scores and
then compare the predicted values to the original y values to compute the
error of prediction values, e. For example, the first y score was 3 with X1 = 2
and X2 = 1. We substitute the X1 and X2 values in the regression equation
and compute a predicted y score of 2.25. The error of prediction is computed as y – this predicted y score or 3 – 2.25 = .75. These computations are

Y102005.indb 436

3/22/10 3:27:33 PM

Appendix A: Introduction to Matrix Operations

437

listed below and are repeated for the remaining y values.
yˆ 1 = .50 + 1( 2) − .25 (1)
yˆ 1 = 2.25
eˆ1 = 3 − 2.25 = .75
yˆ 2 = .50 + 1( 3) − .25 (5)
yˆ 2 = 2.25
eˆ2 = 2 − 2.25 = −.25
yˆ 3 = .50 + 1( 5) − .25 (3)
yˆ 3 = 4.75
eˆ3 = 4 − 4.75 = −.75
yˆ 4 = .50 + 1( 7) − .25 (6)
yˆ 4 = 6.00
eˆ4 = 5 − 6 = −1.00
yˆ 5 = .50 + 1( 8) − .25 (7)
yˆ 5 = 6.75
eˆ5 = 8 − 6.75 = 1.25
The regression equation is: ŷ i = .50 + 1.0X1 − .25 X 2
We can now place the Y values, X values, regression weights, and error
terms back into the matrices to yield a complete solution for the Y values.
Notice that the error term vector should sum to zero (0.0). Also notice that
each y value is uniquely composed of an intercept term (.50), a regression
weight (1.0) times an X1 value, a regression weight (−.25) times an X2 value,
and a residual error, e.g., the first y value of 3 = .5 + 1.0(2) −.25 (1) + .75.
3 

 

2 

 

 4  = .5 + 1.0 
 

5 

 

8 

 


Y102005.indb 437


2 



3 


5  − .25 



7 



8 


1  .75 
 

5   −.25 
 

3  +  −.75 
 

6   −1.00 
 

7  1.25 

3/22/10 3:27:34 PM

Appendix B: Statistical Tables

439

Y102005.indb 439

3/22/10 3:27:35 PM

440

Appendix B: Statistical Tables

Table A.1
Areas under the Normal Curve (z-scores)
Second Decimal Place in z
z

.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

.0
.1
.2
.3
.4

.0000
.0398
.0793
.1179
.1554

.0040
.0438
.0832
.1217
.1591

.0080
.0478
.0871
.1255
.1628

.0120
.0517
.0910
.1293
.1664

.0160
.0557
.0948
.1331
.1700

.0199
.0596
.0987
.1368
.1736

.0239
.0636
.1026
.1406
.1772

.0279
.0675
.1064
.1443
.1808

.0319
.0714
.1103
.1480
.1844

.0359
.0753
.1141
.1517
.1879

.5
.6
.7
.8
.9

.1915
.2257
.2580
.2881
.3159

.1950
.2291
.2611
.2910
.3186

.1985
.2324
.2642
.2939
.3212

.2019
.2357
.2673
.2967
.3238

.2054
.2389
.2704
.2995
.3264

.2088
.2422
.2734
.3023
.3289

.2123
.2454
.2764
.3051
.3315

.2157
.2486
.2794
.3078
.3340

.2190
.2517
.2823
.3106
.3365

.2224
.2549
.2852
.3133
.3389

1.0
1.1
1.2
1.3
1.4

.3413
.3643
.3849
.4032
.4192

.3438
.3665
.3869
.4049
.4207

.3461
.3686
.3888
.4066
.4222

.3485
.3708
.3907
.4082
.4236

.3508
.3729
.3925
.4099
.4251

.3531
.3749
.3944
.4115
.4265

.3554
.3770
.3962
.4131
.4279

.3577
.3790
.3980
.4147
.4292

.3599
.3810
.3997
.4162
.4306

.3621
.3830
.4015
.4177
.4319

1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4

.4332
.4452
.4554
.4641
.4713
.4772
.4821
.4861
.4893
.4918

.4345
.4463
.4564
.4649
.4719
.4778
.4826
.4826
.4896
.4920

.4357
.4474
.4573
.4656
.4726
.4783
.4830
.4868
.4898
.4922

.4793
.4484
.4582
.4664
.4732
.4788
.4834
.4871
.4901
.4925

.4382
.4495
.4591
.4671
.4738
.4793
.4838
.4875
.4904
.4927

.4394
.4505
.4599
.4678
.4744
.4798
.4842
.4878
.4906
.4929

.4406
.4515
.4608
.4686
.4750
.4803
.4846
.4881
.4909
.4931

.4418
.4525
.4616
.4693
.4756
.4808
.4850
.4884
.4911
.4932

.4429
.4535
.4625
.4699
.4761
.4812
.4854
.4887
.4913
.4934

.4441
.4545
.4633
.4706
.4767
.4817
.4857
.4890
.4916
.4936

2.5
2.6
2.7
2.8
2.9

.4938
.4953
.4965
.4974
.4981

.4940
.4955
.4966
.4975
.4982

.4941
.4956
.4967
.4976
.4982

.4943
.4957
.4968
.4977
.4983

.4945
.4959
.4969
.4977
.4984

.4946
.4960
.4970
.4978
.4984

.4948
.4961
.4971
.4979
.4985

.4949
.4962
.4972
.4979
.4985

.4951
.4963
.4973
.4980
.4986

.4952
.4964
.4974
.4981
.4986

3.0
3.1
3.2
3.3
3.4

.4987
.4990
.4993
.4995
.4997

.4987
.4991
.4993
.4995
.4997

.4987
.4991
.4994
.4995
.4997

.4988
.4991
.4994
.4996
.4997

.4988
.4992
.4994
.4996
.4997

.4989
.4922
.4994
.4996
.4997

.4989
.4992
.4994
.4996
.4997

.4989
.4992
.4995
.4996
.4997

.4990
.4993
.4995
.4996
.4997

.4990
.4993
.4995
.4997
.4998

3.5
4.0
4.5
5.0

.4998
.49997
.499997
.4999997

Y102005.indb 440

3/22/10 3:27:36 PM

441

Appendix B: Statistical Tables

Table A.2
Distribution of t for Given Probability Levels
Level of Significance for One-Tailed Test
.10

.05

.025

.01

.005

.0005

Level of Significance for Two-Tailed Test
df

.20

.10

1
2
3
4
5

3.078
1.886
1.638
1.533
1.476

6.314
2.920
2.353
2.132
2.015

6
7
8
9
10

1.440
1.415
1.397
1.383
1.372

11
12
13
14
15

.02

.01

12.706
4.303
3.182
2.776
2.571

31.821
6.965
4.541
3.747
3.365

63.657
9.925
5.841
4.604
4.032

636.619
31.598
12.941
8.610
6.859

1.943
1.895
1.860
1.833
1.812

2.447
2.365
2.306
2.262
2.228

3.143
2.998
2.896
2.821
2.764

3.707
3.499
3.355
3.250
3.169

5.959
5.405
5.041
4.781
4.587

1.363
1.356
1.350
1.345
1.341

1.796
1.782
1.771
1.761
1.753

2.201
2.179
2.160
2.145
2.131

2.718
2.681
2.650
2.624
2.602

3.106
3.055
3.012
2.977
2.947

4.437
4.318
4.221
4.140
4.073

16
17
18
19
20

1.337
1.333
1.330
1.328
1.325

1.746
1.740
1.734
1.729
1.725

2.120
2.110
2.101
2.093
2.086

2.583
2.567
2.552
2.539
2.528

2.921
2.898
2.878
2.861
2.845

4.015
3.965
3.992
3.883
3.850

21
22
23
24
25

1.323
1.321
1.319
1.318
1.316

1.721
1.717
1.714
1.711
1.708

2.080
2.074
2.069
2.064
2.060

2.518
2.508
2.500
2.492
2.485

2.831
2.819
2.807
2.797
2.787

3.819
3.792
3.767
3.745
3.725

26
27
28
29
30

1.315
1.314
1.313
1.311
1.310

1.706
1.703
1.701
1.699
1.697

2.056
2.052
2.048
2.045
2.042

2.479
2.473
2.467
2.462
2.457

2.779
2.771
2.763
2.756
2.750

3.707
3.690
3.674
3.659
3.646

40
60
120
∞

1.303
1.296
1.289
1.282

1.684
1.671
1.658
1.645

2.021
2.000
1.980
1.960

2.423
2.390
2.358
2.326

2.704
2.660
2.617
2.576

3.551
3.460
3.373
3.291

Y102005.indb 441

.05

.001

3/22/10 3:27:36 PM

442

Appendix B: Statistical Tables

Table A.3
Distribution of r for Given Probability Levels
Level of Significance for One-Tailed Test
.05

.025

.01

.005

Level of Significance for Two-Tailed Test
df

.10

.05

.02

.01

1
2
3
4
5

.988
.900
.805
.729
.669

.997
.950
.878
.811
.754

.9995
.980
.934
.882
.833

.9999
.990
.959
.917
.874

6
7
8
9
10

.622
.582
.540
.521
.497

.707
.666
.632
.602
.576

.789
.750
.716
.685
.658

.834
.798
.765
.735
.708

11
12
13
14
15

.576
.458
.441
.426
.412

.553
.532
.514
.497
.482

.634
.612
.592
.574
.558

.684
.661
.641
.623
.606

16
17
18
19
20

.400
.389
.378
.369
.360

.468
.456
.444
.433
.423

.542
.528
.516
.503
.492

.590
.575
.561
.549
.537

21
22
23
24
25

.352
.344
.337
.330
.323

.413
.404
.396
.388
.381

.482
.472
.462
.453
.445

.526
.515
.505
.496
.487

26
27
28
29
30
35
40
45
50
60

.317
.311
.306
.301
.296
.275
.257
.243
.231
.211

.374
.367
.361
.355
.349
.325
.304
.288
.273
.250

.437
.430
.423
.416
.409
.381
.358
.338
.322
.295

.479
.471
.463
.486
.449
.418
.393
.372
.354
.325

70
80
90
100

.195
.183
.173
.164

.232
.217
.205
.195

.274
.256
.242
.230

.303
.283
.267
.254

Y102005.indb 442

3/22/10 3:27:36 PM

Y102005.indb 443

.99

.00016
.0201
.115
.297
.554

.872
1.239
1.646
2.088
2.558

3.053
3.571
4.107
4.660
5.229

5.812
6.408
7.015
7.633
8.260
8.897
9.542
10.196
10.856
11.524

1
2
3
4
5

6
7
8
9
10

11
12
13
14
15

16
17
18
19
20
21
22
23
24
25

6.614
7.255
7.906
8.567
9.237
9.915
10.600
11.293
11.992
12.697

3.609
4.178
4.765
5.368
5.985

1.134
1.564
2.032
2.532
3.059

.00663
.0404
.185
.429
.752

.98

7.962
8.672
9.390
10.117
10.851
11.591
12.338
13.091
13.848
14.611

4.575
5.226
5.892
6.571
7.261

1.635
2.167
2.733
3.325
3.940

.00393
.103
.352
.711
1.145

.95

9.312
10.085
10.865
11.651
12.443
13.240
14.041
14.848
15.659
16.473

5.578
6.304
7.042
7.790
8.547

2.204
2.833
3.490
4.168
4.865

.0158
.211
.584
1.064
1.610

.90

11.152
12.002
12.857
13.716
14.578
15.445
16.314
17.187
18.062
18.940

6.989
7.807
8.634
9.467
10.307

3.070
3.822
4.594
5.380
6.179

.0642
.446
1.005
1.649
2.343

.80

12.624
13.531
14.440
15.352
16.266
17.182
18.101
19.021
19.943
20.867

8.148
9.034
9.926
10.821
11.721

3.828
4.671
5.527
6.393
7.267

.148
.713
1.424
2.195
3.000

.70

Distribution of Chi-Square for Given Probability Levels

Table A.4

15.338
16.338
17.338
18.338
19.337
20.337
21.337
22.337
23.337
24.337

10.341
11.340
12.340
13.339
14.339

5.348
6.346
7.344
8.343
9.342

.455
1.386
2.366
3.357
4.351

.50

18.418
19.511
20.601
21.689
22.775
23.858
24.939
26.018
27.096
28.172

12.899
14.011
15.119
16.222
17.322

7.231
8.383
9.524
10.656
11.781

1.074
2.408
3.665
4.878
6.064

.30

Probability

20.465
21.615
22.760
23.900
25.038
26.171
27.301
28.429
29.553
30.675

14.631
15.812
16.985
18.151
19.311

8.558
9.803
11.030
12.242
13.442

1.642
3.219
4.642
5.989
7.289

.20

23.542
24.769
25.989
27.204
28.412
29.615
30.813
32.007
33.196
34.382

17.275
18.549
19.812
21.064
22.307

10.645
12.017
13.362
14.684
15.987

2.706
4.605
6.251
7.779
9.236

.10

26.296
27.587
28.869
30.144
31.410
32.671
33.924
35.172
36.415
37.652

19.675
21.026
22.362
23.685
24.996

12.592
14.067
15.507
16.919
18.307

3.841
5.991
7.815
9.488
11.070

.05

29.633
30.995
32.346
33.687
35.020
36.343
37.659
38.968
40.270
41.566

22.618
24.054
25.472
26.873
28.259

15.033
16.622
18.168
19.679
21.161

5.412
7.824
9.837
11.668
13.388

.02

32.000
33.409
34.805
36.191
37.566
38.932
40.289
41.638
42.980
44.314

24.725
26.217
27.688
29.141
30.578

16.812
18.475
20.090
21.666
23.209

6.635
9.210
11.345
13.277
15.086

.01

(continued)

39.252
40.790
42.312
43.820
45.315
46.797
48.268
49.728
51.179
52.620

31.264
32.909
34.528
36.123
37.697

22.457
24.322
26.125
27.877
29.588

10.827
13.815
16.266
18.467
20.515

.001

Appendix B: Statistical Tables
443

3/22/10 3:27:36 PM

Y102005.indb 444

23.650
25.148
26.657
28.177
29.707

31.246
32.793
34.350
35.913
37.485

39.063
40.649
42.240
43.838
45.442

42
44
46
48
50

52
54
56
58
60

62
64
66
68
70

.98

41.327
42.960
44.599
46.244
47.893

33.256
34.856
36.464
38.078
39.699

25.383
26.939
28.504
30.080
31.664

17.783
19.275
20.783
22.304
23.838

13.409
14.125
14.847
15.574
16.306

.95

44.889
46.595
48.305
50.020
51.739

36.437
38.116
39.801
41.492
43.188

28.144
29.787
31.439
33.098
34.764

20.072
21.664
23.269
24.884
26.509

15.379
16.151
16.928
17.708
18.493

.90

48.226
49.996
51.770
53.548
55.329

39.433
41.183
42.937
44.696
46.459

30.765
32.487
34.215
35.949
37.689

22.271
23.952
25.643
27.343
29.051

17.292
18.114
18.939
19.768
20.599

.80

52.487
54.336
56.188
58.042
59.898

43.281
45.117
46.955
48.797
50.641

34.147
35.974
37.795
39.621
41.449

25.148
26.938
28.735
30.537
32.345

19.820
20.703
21.588
22.475
23.364

.70

55.714
57.620
59.527
61.436
63.346

46.209
48.106
50.005
51.906
53.809

36.755
38.641
40.529
42.420
44.313

27.373
29.242
31.115
32.992
34.872

21.792
22.719
23.647
24.577
25.508

61.335
63.335
65.335
67.335
69.335

51.335
53.335
55.335
57.335
59.335

41.335
43.335
45.335
47.335
49.335

31.336
33.336
35.336
37.335
39.335

25.336
26.336
27.336
28.336
29.336

.50

67.322
69.416
71.508
73.600
75.689

56.827
58.930
61.031
63.129
65.227

46.282
48.396
50.507
52.616
54.723

35.665
37.795
39.922
42.045
44.165

29.246
30.319
31.391
32.461
33.530

.30

Probability
.20

71.125
73.276
75.424
77.571
79.715

60.332
62.496
64.658
66.816
68.972

49.456
51.639
53.818
55.993
58.164

38.466
40.676
42.879
45.076
47.269

31.795
32.912
34.027
35.139
36.250

.10

76.630
78.860
81.085
83.308
85.527

65.422
67.673
69.919
72.160
74.397

54.090
56.369
58.641
60.907
63.167

42.585
44.903
47.212
49.513
51.805

35.563
36.741
37.916
39.087
40.256

.05

81.381
83.675
85.965
88.250
90.531

69.832
72.153
74.468
76.778
79.082

58.124
60.481
62.830
65.171
67.505

46.194
48.602
50.999
53.384
55.759

38.885
40.113
41.337
42.557
43.773

.02

86.953
89.320
91.681
94.037
96.388

75.021
77.422
79.815
82.201
84.580

62.892
65.337
67.771
70.197
72.613

50.487
52.995
55.489
57.969
60.436

42.856
44.140
45.419
46.693
47.962

.01

90.802
93.217
95.626
98.028
100.425

78.616
81.069
83.513
85.950
88.379

66.206
68.710
71.201
73.683
76.154

53.486
56.061
58.619
61.162
63.691

45.642
46.963
48.278
49.588
50.892

102.166
104.716
107.258
109.791
112.317

89.272
91.872
94.461
97.039
99.607

76.084
78.750
81.400
84.037
86.661

62.487
65.247
67.985
70.703
73.402

54.052
55.476
56.893
58.302
59.703

.001

Note. For larger values of df, the expression √ (X2)2 − √ 2df – 1 may be used as a normal deviate with unit variance, remembering that the probability
for X2 corresponds with that of a single tail of the normal curve.

16.362
17.789
19.233
20.691
22.164

32
34
36
38
40

.99

12.198
12.879
13.565
14.256
14.953

26
27
28
29
30

Distribution of Chi-Square for Given Probability Levels

Table A.4 (Continued)

444
Appendix B: Statistical Tables

3/22/10 3:27:37 PM

Y102005.indb 445

120

∞

6.61
5.99
5.59
5.32
5.12

4.96
4.84
4.75
4.67
4.60

4.54
4.49
4.45
4.41
4.38

10
11
12
13
14

15
16
17
18
19

3.68
3.63
3.59
3.55
3.52

4.10
3.98
3.89
3.81
3.74

5.79
5.14
4.74
4.46
4.26

3.29
3.24
3.20
3.16
3.13

3.71
3.59
3.49
3.41
3.34

5.41
4.76
4.35
4.07
3.86

3.06
3.01
2.96
2.93
2.90

3.48
3.36
3.26
3.18
3.11

5.19
4.53
4.12
3.84
3.63

2.90
2.85
2.81
2.77
2.74

3.33
3.20
3.11
3.03
2.96

5.05
4.39
3.97
3.69
3.48

2.79
2.74
2.70
2.66
2.63

3.22
3.09
3.00
2.92
2.85

4.95
4.28
3.87
3.58
3.37

2.71
2.66
2.61
2.58
2.54

3.14
3.01
2.91
2.83
2.76

4.88
4.21
3.79
3.50
3.29

2.64
2.59
2.55
2.51
2.48

3.07
2.95
2.85
2.77
2.70

4.82
4.15
3.73
3.44
3.23

2.59
2.54
2.49
2.46
2.42

3.02
2.90
2.80
2.71
2.65

4.77
4.10
3.68
3.39
3.18

2.54
2.49
2.45
2.41
2.38

2.98
2.85
2.75
2.67
2.60

4.74
4.06
3.64
3.35
3.14

2.48
2.42
2.38
2.34
2.31

2.91
2.79
2.69
2.60
2.53

4.68
4.00
3.57
3.28
3.07

2.40
2.35
2.31
2.27
2.23

2.85
2.72
2.62
2.53
2.46

4.62
3.94
3.51
3.22
3.01

2.33
2.28
2.23
2.19
2.16

2.77
2.65
2.54
2.46
2.39

4.56
3.87
3.44
3.15
2.94

2.29
2.24
2.19
2.15
2.11

2.74
2.61
2.51
2.42
2.35

4.53
3.84
3.41
3.12
2.90

2.25
2.19
2.15
2.11
2.07

2.70
2.57
2.47
2.38
2.31

4.50
3.81
3.38
3.08
2.86

2.20
2.1
2.10
2.06
2.03

2.66
2.53
2.43
2.34
2.27

4.46
3.77
3.34
3.04
2.83

2.16
2.11
2.06
2.02
1.98

2.62
2.49
2.38
2.30
2.22

4.43
3.74
3.30
3.01
2.79

2.07
2.01
1.96
1.92
1.88

2.54
2.40
2.30
2.21
2.13

4.36
3.67
3.23
2.93
2.71

(continued)

2.11
2.06
2.01
1.97
1.93

2.58
2.45
2.34
2.25
2.18

4.40
3.70
3.27
2.97
2.75

161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5 241.9 243.9 245.9 248.0 249.1 250.1 251.1 252.2 253.3 254.3
18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.49 19.41 19.43 19.45 19.45 19.46 19.47 19.48 19.49 19.50
10.13
9.55
9.28
9.12
9.01
8.94
8.89
8.85
8.81
8.79
8.74
8.70
8.66
8.64
8.62
8.59
8.57
8.55
8.53
7.71
6.94
6.59
6.39
6.26
6.15
6.09
6.04
6.00
5.96
5.91
5.86
5.80
5.77
5.75
5.72
5.69
5.66
5.63

5
6
7
8
9

1
2
3
4

df1 df2

The F-Distribution for Given Probability Levels (.05 Level)

Table A.5

Appendix B: Statistical Tables
445

3/22/10 3:27:37 PM

Y102005.indb 446

4.35
4.32
4.30
4.28
4.26

4.24
4.23
4.21
4.20
4.18

4.17
4.08
4.00
3.92
3.84

df1 df2

20
21
22
23
24

25
26
27
28
29

30
40
60
120
∞

3.32
3.23
3.15
3.07
3.00

3.39
3.37
3.35
3.34
3.33

3.49
3.47
3.44
3.42
3.40

2.92
2.84
2.76
2.68
2.60

2.99
2.98
2.96
2.95
2.93

3.10
3.07
3.05
3.03
3.01

2.69
2.61
2.53
2.45
2.37

2.76
2.74
2.73
2.71
2.70

2.87
2.84
2.82
2.80
2.78

2.53
2.45
2.37
2.29
2.21

2.60
2.59
2.57
2.56
2.55

2.71
2.68
2.66
2.64
2.62

2.42
2.34
2.25
2.17
2.10

2.49
2.47
2.46
2.45
2.43

2.60
2.57
2.55
2.53
2.51

2.33
2.25
2.17
2.09
2.01

2.40
2.39
2.37
2.36
2.35

2.51
2.49
2.46
2.44
2.42

2.27
2.18
2.10
2.02
1.94

2.34
2.32
2.31
2.29
2.28

2.45
2.42
2.40
2.37
2.36

The F-Distribution for Given Probability Levels (.05 Level)

Table A.5 (Continued)
9

2.21
2.12
2.04
1.96
1.88

2.28
2.27
2.25
2.24
2.22

2.39
2.37
2.34
2.32
2.30

2.16
2.08
1.99
1.91
1.83

2.24
2.22
2.20
2.19
2.18

2.35
2.32
2.30
2.27
2.25

2.09
2.00
1.92
1.83
1.75

2.16
2.15
2.13
2.12
2.10

2.28
2.25
2.23
2.20
2.18

2.01
1.92
1.84
1.75
1.67

2.09
2.07
2.06
2.04
2.03

2.20
2.18
2.15
2.13
2.11

1.93
1.84
1.75
1.66
1.57

2.01
1.99
1.97
1.96
1.94

2.12
2.10
2.07
2.05
2.03

1.89
1.79
1.70
1.61
1.52

1.96
1.95
1.93
1.91
1.90

2.08
2.05
2.03
2.01
1.98

1.84
1.74
1.65
1.55
1.46

1.92
1.90
1.88
1.87
1.85

2.04
2.01
1.98
1.96
1.94

1.79
1.69
1.59
1.50
1.39

1.87
1.85
1.84
1.82
1.81

1.99
1.96
1.94
1.91
1.89

1.74
1.64
1.53
1.43
1.32

1.82
1.80
1.79
1.77
1.75

1.95
1.92
1.89
1.86
1.84

1.68
1.58
1.47
1.35
1.22

1.77
1.75
1.73
1.71
1.70

1.90
1.87
1.84
1.81
1.79

120

∞

1.62
1.51
1.39
1.25
1.00

1.71
1.69
1.67
1.65
1.64

1.84
1.81
1.78
1.76
1.73

446
Appendix B: Statistical Tables

3/22/10 3:27:38 PM

Y102005.indb 447

120

16.26
13.75
12.25
11.26
10.56

10.04
9.65
9.33
9.07
8.86

8.68
8.53
8.40
8.29
8.18

5
6
7
8
9

10
11
12
13
14

15
16
17
18
19

6.36
6.23
6.11
6.01
5.93

7.56
7.21
6.93
6.70
6.51

5.42
5.29
5.18
5.09
5.01

6.55
6.22
5.95
5.74
5.56

13.27 12.06
10.92 9.78
9.55 8.45
8.65 7.59
8.02 6.99

4.89
4.77
4.67
4.58
4.50

5.99
5.67
5.41
5.21
5.04

11.39
9.15
7.85
7.01
6.42

4.56
4.44
4.34
4.25
4.17

5.64
5.32
5.06
4.86
4.69

10.97
8.75
7.46
6.63
6.06

4.32
4.20
4.10
4.01
3.94

5.39
5.07
4.82
4.62
4.46

10.67
8.47
7.19
6.37
5.80

4.14
4.03
3.93
3.84
3.77

5.20
4.89
4.64
4.44
4.28

10.46
8.26
6.99
6.18
5.61

4.00
3.89
3.79
3.71
3.63

5.06
4.74
4.50
4.30
4.14

10.29
8.10
6.84
6.03
5.47

3.89
3.78
3.68
3.60
3.52

4.94
4.63
4.39
4.19
4.03

10.16
7.98
6.72
5.91
5.35

3.80
3.69
3.59
3.51
3.43

4.85
4.54
4.30
4.10
3.94

10.05
7.87
6.62
5.81
5.26

3.67
3.55
3.46
3.37
3.30

4.71
4.40
4.16
3.96
3.80

9.89
7.72
6.47
5.67
5.11

3.52
3.41
3.31
3.23
3.15

4.56
4.25
4.01
3.82
3.66

9.72
7.56
6.31
5.52
4.96

3.37
3.26
3.16
3.08
3.00

4.41
4.10
3.86
3.66
3.51

9.55
7.40
6.16
5.36
4.81

3.29
3.18
3.08
3.00
2.92

4.33
4.02
3.78
3.59
3.43

9.47
7.31
6.07
5.28
4.73

3.21
3.10
3.00
2.92
2.84

4.25
3.94
3.70
3.51
3.35

9.38
7.23
5.99
5.20
4.65

3.13
3.02
2.92
2.84
2.76

4.17
3.86
3.62
3.43
3.27

9.29
7.14
5.91
5.12
4.57

3.05
2.93
2.83
2.75
2.67

4.08
3.78
3.54
3.34
3.18

9.20
7.06
5.82
5.03
4.48

2.87
2.75
2.65
2.57
2.49

3.91
3.60
3.36
3.17
3.00

9.02
6.88
5.65
4.86
4.31

6366
99.50
26.13
13.46

∞

(continued)

2.96
2.84
2.75
2.66
2.58

4.00
3.69
3.45
3.25
3.09

9.11
6.97
5.74
4.95
4.40

1 4052 4999.5 5403 5625 5764 5859 5928 5982 6022 6056 6106 6157 6209 6235 6261 6287 6313 6339
2 98.5
99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39 99.40 99.42 99.43 99.45 99.46 99.47 99.47 99.48 99.49
3 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.25 27.23 27.05 26.87 26.69 26.60 26.50 26.41 26.32 26.22
4 21.20 18.00 16.69 15.98
5.52 15.21 14.98 14.80 14.66 14.55 14.37 14.20 14.02 13.93 13.84 13.75 13.65 13.56

df1
df2

The F Distribution for Given Probability Levels (.01 Level)

Table A.6

Appendix B: Statistical Tables
447

3/22/10 3:27:38 PM

Y102005.indb 448

8.10
8.02
7.95
7.88
7.82

7.77
7.72
7.68
7.64
7.60

7.56
7.31
7.08
6.85
6.63

df1
df2

20
21
22
23
24

25
26
27
28
29

30
40
60
120
∞

5.39
5.18
4.98
4.79
4.61

5.57
5.53
5.49
5.45
5.42

5.85
5.78
5.72
5.66
5.61

4.51
4.31
4.13
3.95
3.78

4.68
4.64
4.60
4.57
4.54

4.94
4.87
4.82
4.76
4.72

4.02
3.83
36.5
3.48
3.32

4.18
4.14
4.11
4.07
4.04

4.43
4.37
4.31
4.26
4.22

3.70
3.51
3.34
3.17
3.02

3.85
3.82
3.78
3.75
3.73

4.10
4.04
3.9
3.94
3.90

3.47
3.29
3.12
2.96
2.80

3.63
3.59
3.56
3.53
3.50

3.87
3.81
3.76
3.71
3.67

3.30
3.12
2.95
2.79
2.64

3.46
3.42
3.39
3.36
3.33

3.70
3.64
3.59
3.54
3.50

3.17
2.99
2.82
2.66
2.51

3.32
3.29
3.26
3.23
3.20

3.56
3.51
3.45
3.41
3.36

The F Distribution for Given Probability Levels (.01 Level)

Table A.6 (Continued)

3.07
2.89
2.72
2.56
2.41

3.22
3.18
3.15
3.12
3.09

3.46
3.40
3.35
3.30
3.26

2.98
2.80
2.63
2.47
2.32

3.13
3.09
3.06
3.03
3.00

3.37
3.31
3.26
3.21
3.17

2.84
2.66
2.50
2.34
2.18

2.99
2.96
2.93
2.90
2.87

3.23
3.17
3.12
3.07
3.03

2.70
2.52
2.35
2.19
2.04

2.85
2.81
2.78
2.75
2.73

3.09
3.03
2.98
2.93
2.89

2.55
2.37
2.20
2.03
1.88

2.70
2.66
2.63
2.60
2.57

2.94
2.88
2.83
2.78
2.74

2.47
2.29
2.12
1.95
1.79

2.62
2.58
2.55
2.52
2.49

2.86
2.80
2.75
2.70
2.66

2.39
2.20
2.03
1.86
1.70

2.54
2.50
2.47
2.44
2.41

2.78
2.72
2.67
2.62
2.58

2.30
2.11
1.94
1.76
1.59

2.45
2.42
2.38
2.35
2.33

2.69
2.64
2.58
2.54
2.49

2.21
2.02
1.84
1.66
1.47

2.36
2.33
2.29
2.26
2.23

2.61
2.55
2.50
2.45
2.40

2.11
1.92
1.73
1.53
1.32

2.27
2.23
2.20
2.17
2.14

2.52
2.46
2.40
2.35
2.31

120

2.01
1.80
1.60
1.38
1.00

2.17
2.13
2.10
2.06
2.03

2.42
2.36
2.31
2.26
2.21

∞

448
Appendix B: Statistical Tables

3/22/10 3:27:39 PM

Answers to Selected Exercises

Chapter 1
1. Define the following terms:
a. Latent variable: an unobserved variable that is not directly measured, but is computed using multiple observed variables.
b. Observed variable: a raw score obtained from a test or measurement instrument on a trait of interest.
c. Dependent variable: a variable that is measured and related to
outcomes, performance, or criterion.
d. Independent variable: a variable that defines mutually exclusive categories (e.g., gender, region, or grade level), or as a continuous variable, and influences a dependent variable.
3. List the reasons why a researcher would conduct structural equation modeling:
a. Researchers are becoming more aware of the need to use multiple observed variables to better understand their area of scientific inquiry.
b. More recognition is given to the validity and reliability of
observed scores from measurement instruments.
c. Structural equation modeling has improved recently, especially the ability to analyze more advanced statistical models.
d. SEM software programs have become increasingly user friendly.

Chapter 2
1. LISREL uses which command to import data sets?
c. File, then Import Data
3. Mark each of the following statements true (T) or false (F).
a. LISREL can deal with missing data.
F
449

Y102005.indb 449

3/22/10 3:27:39 PM

450

Answers to Selected Exercises

b. PRELIS can deal with missing data.
T
c. LISREL can compute descriptive statistics.
d. PRELIS can compute descriptive statistics.

T
T

Chapter 3
1. Partial and part correlations:
r 12.3 =

.6 − (.7 )(.4)
2

[1 − (.7 ) ][1 − (.4) ]

r 1( 2.3 ) =

.6 − (.7 ) (.4)
2

[1 − (.4) ]

= .49

= .35.

3. A meaningful theoretical relationship should be plausible
given that:
a. Variables logically precede each other in time.
b. Variables covary or correlate together as expected.
c. Other influences or “causes” are controlled.
d. Variables should be measured on at least an interval level.
e. Changes in a preceding variable should affect variables that
follow, either directly or indirectly.

Chapter 4
1. Model specification: developing a theoretical model to test, based
on all of the relevant theory, research, and information available.
3. Model estimation: obtaining estimates for each of the parameters
specified in the model that produced the implied population covariance matrix Σ. The intent is to obtain parameter estimates that
yield a matrix Σ as close as possible to S, our sample covariance
matrix of the observed or indicator variables. When elements in
the matrix S minus the elements in the matrix Σ equal zero (S – Σ
= 0), then c2 = 0 indicating a perfect model fit to the data, and all
values in S are equal to values in Σ.

Y102005.indb 450

3/22/10 3:27:40 PM

Answers to Selected Exercises

451

5. Model modification: changing the initial implied model and
retesting the global fit and individual parameters in the new
respecified model. To determine how to modify the model,
there are a number of procedures available to guide the adding
or dropping of paths in the model so that alternative models can
be tested.
7. How many distinct values are in a variance–covariance matrix
for the following variables {hint: [p(p+1)/2}?
a. Five variables = 15 distinct values
b. Ten variables = 55 distinct values

Chapter 5
1. Define confirmatory models, alternative models, and modelgenerating approaches.
In confirmatory models, a researcher can hypothesize a specific theoretical model, gather data, and then test whether the data fit
the model.
In alternative models, a researcher specifies different models to see
which model fits the sample data the best. A researcher usually conducts a chi-square difference test.
In model generating, a researcher specifies an initial model, then
uses modification indices to modify and retest the model to
obtain a better fit to the sample data.
3. Calculate the following fit indices for the model analysis in
Figure 5.1:
GFI = 1 – [c2model/c2null] = .97
NFI = (c2null − c2model)/c2null = .97
RFI = 1 – [(c2model/dfmodel)/(c2null /dfnull)] = .94
IFI = (c2null − c2model)/(c2null − dfmodel) = .98
TLI = [(c2null/dfnull) − (c2model/dfmodel)]/[(c2null/dfnull) − 1] = .96
CFI = 1 – [(c2model − dfmodel)/(c2null − dfnull)] = .98
Model AIC = c2model + 2q = 50.41
Null AIC = c2 null + 2q = 747.80
2
RMSEA = [ χ Model
− df Model ]/[( N − 1)df Model ] = 0.083

Y102005.indb 451

3/22/10 3:27:40 PM

452

Answers to Selected Exercises

5. What steps should a researcher take in examining parameter estimates in a model?
A researcher should examine the sign of the parameter estimate,
whether the value of the parameter estimate is within a reasonable range of values, and test the parameter for significance.
7. How are structural equation models affected by sample size and
power considerations?
Several factors affect determining the appropriate sample size
and power, including model complexity, distribution of variables,
missing data, reliability, and variance–covariance of variables. If
variables are normally distributed with no missing data, samples
sizes less than 500 should yield power = .80 and satisfy Hoelter’s
CN criterion. SAS, SPSS, G*Power 3, and other software programs
can be used to determine power and sample size.
9. What new approaches are available to help researchers identify
the best model?
The expected parameter change value has been added to LISREL
output. Tabu and optimization algorithms have been proposed to
identify the best model fit with the sample variance–covariance
matrix.
11. Use G*Power 3 to calculate power for modified model with
alpha = .05 and NCP = 6.3496 at df = 1, df = 2, and df = 3 levels
of model complexity. What happens to power when degrees of
freedom increases?
Power decreases as the degrees of freedom increases (power = .73,
df = 1; power = .63, df = 2, and power = .56, df = 3).

Y102005.indb 452

3/22/10 3:27:40 PM

453

Answers to Selected Exercises

Critical χ2 = 3.84146
0.6
0.4
0.2
0

Y102005.indb 453

β
0

α
2

3/22/10 3:27:41 PM

454

Answers to Selected Exercises

Critical χ2 = 5.99146

0.5
0.4
0.3
0.2
β

0.1
0

Y102005.indb 454

α
0

3/22/10 3:27:42 PM

455

Answers to Selected Exercises

Critical χ2 = 7.81473
0.2
0.15
0.1
0.05
0

Y102005.indb 455

β
0

3/22/10 3:27:43 PM

456

Answers to Selected Exercises

Chapter 6
1. The following LISREL–SIMPLIS program is run to analyze the
theoretical regression model for predicting gross national product (GNP) from knowledge of labor, capital, and time:
Regression of GNP
Observed variables: GNP LABOR CAPITAL TIME
Covariance matrix:
4256.530
449.016 52.984
1535.097 139.449 1114.447
537.482 53.291 170.024 73.747
Sample size: 23
Equation: GNP = LABOR CAPITAL TIME
Number of decimals = 3
Path diagram
End of problem

Chapter 7
1. LISREL PROGRAM
Achievement path model
Observed variables: Ach Inc Abl Asp
Covariance matrix:
25.500
20.500 38.100
22.480 24.200 42.750
16.275 13.600 13.500 17.000
Sample size: 100
Relationships
Asp = Inc Abl
Ach = Inc Abl Asp
Print residuals
Options: ND = 3
Path diagram
End of problem

Partial LISREL Output
LISREL Estimates (Maximum Likelihood)
Structural Equations

Y102005.indb 456

3/22/10 3:27:43 PM

457

Answers to Selected Exercises

Ach = 0.645*Asp + 0.161*Inc + 0.231*Abl, Errorvar. = 6.507,
R² = 0.745
(0.0771) (0.0557) (0.0514) (0.934)
8.366
2.892 4.497 6.964
Asp = 0.244*Inc + 0.178*Abl, Errorvar. = 11.282, R² = 0.336
(0.0690) (0.0652) (1.620)
3.537 2.724 6.964
Covariance Matrix of Independent Variables
Inc
-------38.100
(5.471)
6.964
24.200
(4.778)
5.065

Inc

Abl

Abl
--------

42.750
(6.139)
6.964

Goodness-of-Fit Statistics
Degrees of Freedom = 0
Minimum Fit Function Chi-Square = 0.00 (P = 1.000)
Normal Theory Weighted Least Squares Chi-Square = 0.00 (P =
1.000)
The model is saturated, the fit is perfect!

Chapter 8
1. The following LISREL–SIMPLIS program was written:
Confirmatory Factor Model Exercise Chapter 8
Observed Variables:
Academic Concept Aspire Degree Prestige Income
Correlation Matrix
1.000
0.487
1.000
0.236
0.206
1.000
0.242
0.179
0.253
1.000
0.163
0.090
0.125
0.481
1.000
0.064
0.040
0.025
0.106
0.136
1.000
Sample Size: 3094
Latent Variables: Motivate SES

Y102005.indb 457

3/22/10 3:27:43 PM

458

Answers to Selected Exercises

Relationships:
Academic - Aspire = Motivate
Degree - Income = SES
Print Residuals
Number of Decimals = 3
Path diagram
End of problem

Results overall suggest a less than acceptable fit:
Normal Theory Weighted Least Squares Chi-Square = 114.115 (P = 0.0)
Degrees of Freedom = 8
Root Mean Square Error of Approximation (RMSEA) = 0.0655
Standardized RMR = 0.0377
Goodness-of-Fit Index (GFI) = 0.988
Consequently, the model modification indices were examined. The largest
decrease in chi-square results from adding an error covariance between
Concept and Academic (boldfaced), thus allowing us to maintain a
hypothesized two-factor model.
The Modification Indices Suggest to Add the
Path to
from
Decrease in Chi-Square
Concept
SES
21.9
Aspire
SES
78.0
Degree
Motivate
16.1
Prestige
Motivate
18.1

New Estimate
−0.14
0.21
0.31
−0.22

The Modification Indices Suggest to Add an Error Covariance
Between
and
Decrease in Chi-Square
New Estimate
Concept
Academic
78.0
0.63
Aspire
Academic
21.9
−0.12
Degree
Aspire
75.3
0.13
Prestige
Concept
8.9
−0.04
Income
Degree
18.1
−0.10
Income
Prestige
16.1
0.07

The following error covariance command line was added.
Let the errors Concept and Academic correlate
The results indicated further model modifications. The largest decrease
in chi-square was determined to occur by adding an error covariance
between Income and Prestige (boldfaced in following text), thus maintaining our hypothesized two-factor confirmatory model.

Y102005.indb 458

3/22/10 3:27:43 PM

459

Answers to Selected Exercises

The Modification Indices Suggest to Add the
Path to
from
Decrease in Chi-Square
Degree
Motivate
20.3
Prestige
Motivate
18.4

New Estimate
0.71
−0.39

The Modification Indices Suggest to Add an Error Covariance
Between
and
Decrease in Chi-Square
New Estimate
Degree
Aspire
10.0
0.09
Prestige
Aspire
8.3
–0.05
Income
Degree
18.4
–0.10
Income
Prestige
20.3
0.08

The following error covariance command line was added.
Let the errors Income and Prestige correlate
The final results indicated a more acceptable level of fit:
Normal Theory Weighted Least Squares Chi-Square = 14.519
(P = 0.0243)
Degrees of Freedom = 6
Root Mean Square Error of Approximation (RMSEA) = 0.0214
Standardized RMR = 0.0123
Goodness-of-Fit Index (GFI) = 0.998
The final LISREL–SIMPLIS program was:
Modified Confirmatory Factor Model - Exercise Chapter 8
Observed Variables:
Academic Concept Aspire Degree Prestige Income
Correlation Matrix
1.000
0.487 1.000
0.236 0.206 1.000
0.242 0.179 0.253 1.000
0.163 0.090 0.125 0.481 1.000
0.064 0.040 0.025 0.106 0.136 1.000
Sample Size: 3094
Latent Variables: Motivate SES
Relationships:
Academic: Aspire = Motivate
Degree: Income = SES
Let the errors concept and Academic correlate
Let the errors Income and Prestige correlate
Print residuals
Number of decimals = 3
Path diagram
End of problem

Y102005.indb 459

3/22/10 3:27:43 PM

460

Answers to Selected Exercises

Chapter 9
1. Diagram two indicator variables X1 and X2 of a latent variable LV.
e_1

X1
LV

e_2

3. Diagram a latent independent variable LIV predicting a latent
dependent variable LDV.
e

LIV

LDV

Chapter 10
1. The following LISREL–SIMPLIS program was written:
Chapter 10 Exercise
Observed variables: ACT CGPA ENTRY SALARY PROMO
Covariance matrix:
1.024
.792 1.077
.567 .537 .852
.445 .424 .518 .670
.434 .389 .475 .545 .716
Sample size: 500
Latent variables: ACAD JOB
Relationships:
ACT = 1*ACAD
CGPA = ACAD
ENTRY = ACAD
SALARY = 1*JOB
PROMO = JOB
JOB = ACAD
Path diagram
End of problem

Y102005.indb 460

3/22/10 3:27:44 PM

461

Answers to Selected Exercises

The chi-square is statistically significant (c2 = 116.29, df = 4, p = .000), so the
modification indices are checked and it is suggested to add an error covariance between the measurement error variances of CGPA and ACT.
The Modification Indices Suggest Adding an Error Covariance
Between
ACT
CGPA
ENTRY
ENTRY
ENTRY

and
SALARY
ACT
SALARY
ACT
CGPA

Decrease in Chi-Square
14.0
113.5
40.8
24.9
23.9

New Estimate
–0.06
0.43
0.10
–0.15
–0.14

The following command line was added:
Let the error covariances between CGPA and ACT correlate

The modified model is acceptable (c2 = 3.04, df = 3, p = .39; RMSEA = .005;
GFI = 1.0). JOB is statistically significantly predicted, R2 = .70, by the following structural equation:
Structural Equations
JOB = 0.91*ACAD, Errorvar.= 0.18 , R² = 0.70
(0.061) (0.027)
15.01 6.59

Chapter 11
1. SPSS and EXCEL matrix input.
SPSS Matrix Input Example

Y102005.indb 461

3/22/10 3:27:44 PM

462

Answers to Selected Exercises

Microsoft Excel Matrix Input Example

Chapter 12
1. Multiple Samples
LISREL–SIMPLIS Program (EX11B.SPL)
Sample 1: Parental Socioeconomic Characteristics
Observed Variables: SOFED SOMED SOFOC FAFED MOMED FAFOC
Covariance Matrix
5.86
3.12 3.32
35.28 23.85 622.09
4.02 2.14 29.42 5.33
2.99 2.55 19.20 3.17 4.64
35.30 26.91 465.62 31.22 23.38 546.01
Sample Size: 80
Latent Variables: Fed Med Foc
SOFED = Fed
SOMED = Med
SOFOC = Foc
FAFED = 1*Fed
MOMED = 1*Med
FAFOC = 1*Foc
Set the Error Covariance between SOMED and SOFED free

Y102005.indb 462

3/22/10 3:27:45 PM

Answers to Selected Exercises

463

Sample 2: Parental Socioeconomic Characteristics
Covariance Matrix
8.20
3.47 4.36
45.65 22.58 611.63
6.39 3.16 44.62 7.32
3.22 3.77 23.47 3.33 4.02
45.58 22.01 548.00 40.99 21.43 585.14
SOFED = Fed
SOMED = Med
SOFOC = Foc
Let the Error Variances of SOFED - SOFOC be free
Set the Error Covariance between SOMED and SOFED free
Sample 3: Parental Socioeconomic Characteristics
Covariance Matrix
5.74
1.35 2.49
39.24 12.73 535.30
4.94 1.65 37.36 5.39
1.67 2.32 15.71 1.85 3.06
40.11 12.94 496.86 38.09 14.91 538.76
SOFED = Fed
SOMED = Med
SOFOC = Foc
Let the Error Variances of SOFED - SOFOC be free
Set the Error Covariance between SOMED and SOFED equal to 0
Path diagram
End of problem
Global Goodness-of-Fit Statistics
Degrees of Freedom = 34
Minimum Fit Function Chi-Square = 52.73 (P = 0.021)
Root Mean Square Error of Approximation (RMSEA) = 0.077
90 Percent Confidence Interval for RMSEA = (0.019; 0.12)
P-Value for Test of Close Fit (RMSEA < 0.05) = 0.00038
Normed Fit Index (NFI) = 0.96
Comparative Fit Index (CFI) = 0.99
Critical N (CN) = 252.98

Y102005.indb 463

3/22/10 3:27:45 PM

464

Answers to Selected Exercises

Chapter 13
1. Multiple Sample Model
The two semesters of data did not have means and standard deviations
on the measures for the regression model, so no means and standard
deviations would be included in the multiple sample LISREL–SIMPLIS
program. (Note: Although two samples are used, we still use the GROUP
command.) The LISREL–SIMPLIS program is:
Predicting Clinical Competence in Nursing
Group 1: Semester 1
Observed variables comp effort learn
Sample size: 250
Correlation matrix
1.0
.25 1.0
.28 .23 1.0
Equation
comp = effort learn
Group 2: Semester 2
Observed variables comp effort learn
Sample size: 205
Correlation matrix
1.0
.21 1.0
.16 .15 1.0
Path diagram
End of problem

Computer Output—Multiple Sample Model
The regression model output indicated a nonsignificant chi-square (chisquare = 1.55, df = 3, p = .67), which implies that the two semesters of sample data had similar regression coefficients. We find that the regression
coefficient of effort predicting comp is .20 compared to .25 and .21, respectively, in the two samples. We also find that the regression coefficient of
learn predicting comp is .19 compared to .28 and .16, respectively, in the
two samples. The correlation between effort and learn is .23 in the common regression model, compared to .23 and .15, respectively, in the two
samples of data. Finally, we see that the R-squared for the common regression model is .19 (1-R-squared = .91). The computer output (not shown)

Y102005.indb 464

3/22/10 3:27:45 PM

465

Answers to Selected Exercises

indicated R-squared = .09 and .085, respectively, for the two regression
equations from the two samples of data.
eﬀort

0.20
comp

0.23

0.91

0.19
learn
Chi-Square = 1.55, df = 3, P-value = 0.67

3. Structured Means Model
The two stacked LISREL–PRELIS programs are:
Group Low Motivation
Observed Variables: Prod1 Prod2 Prod3 Prod4 Prod5 Prod6
Correlation Matrix
1.00
.64 1.00
.78 .73 1.00
.68 .63
.69 1.00
.43 .55
.50
.59 1.00
.65 .63
.67
.81
.60 1.00
Means 4.27 5.02
4.48 4.69 4.53 4.66
Sample Size: 300
Latent Variables: City1 City2
Relationships:
Prod1 = CONST + 1*City1
Prod2 = CONST + City1
Prod3 = CONST + City1
Prod4 = CONST + 1*City2
Prod5 = CONST + City2
Prod6 = CONST + City2
Group High Motivation:
Correlation Matrix
1.00
.72 1.00
.76 .74 1.00
.51 .46
.57 1.00
.32 .33
.39
.40 1.00
.54 .45
.60
.73
.45 1.00
Means 14.35 14.93 14.59 14.86 14.71 14.74

Y102005.indb 465

3/22/10 3:27:45 PM

466

Answers to Selected Exercises

Sample size: 300
Relationships:
City1 = CONST
City2 = CONST
Path diagram
End of problem

The first thing you should check is the individual group and combined
group model-fit statistics. They were:
Group Goodness-of-Fit Statistics: Low Motivation
Contribution to Chi-Square = 52.92
Root Mean Square Residual (RMR) = 0.11
Goodness-of-Fit Index (GFI) = 0.94
Group Goodness-of-Fit Statistics: High Motivation
Contribution to Chi-Square = 52.06
Root Mean Square Residual (RMR) = 0.13
Goodness-of-Fit Index (GFI) = 0.94
Global Goodness-of-Fit Statistics
Degrees of Freedom = 24
Minimum Fit Function Chi-Square = 104.98 (P = 0.00)
Root Mean Square Error of Approximation (RMSEA) = 0.11
90% Confidence Interval for RMSEA = (0.089; 0.13)
P-Value for Test of Close Fit (RMSEA < 0.05) = 0.00
Comparative Fit Index (CFI) = 0.97
These values are adequate, but modification indices were indicated and
are suggested to yield a better model fit before proceeding with a test of
latent variable mean differences.
The following command lines should be added to the LISREL–
SIMPLIS program to allow observed variable error variance to be
estimated, estimate latent variable variance, and allow the two latent
variables to correlate:
Set the Error Variances of Prod1 - Prod6 free
Set the Variances of City1 - City2 free
Set the Covariance between City1 and City2 free

Y102005.indb 466

3/22/10 3:27:46 PM

467

Answers to Selected Exercises

The final Structured Means Model with parameter estimates is:
0.28

Prod1
0.86

0.34

Prod2

0.83

City1

0.89
0.20

Prod3
0.90

0.21

Prod4
0.85

0.43

Prod5

0.71

City2

0.85
0.23

Prod6

The Structured Means Model is testing the mean latent variable difference, which is indicated by the Mean Vector of Independent Variables.
Results are interpreted based on the knowledge that the mean latent
value on City1 (Los Angeles) and City2 (Chicago) are set to zero (0) in
the first group (low motivation), so the values reported here are going to
indicate that the second group (high motivation) was either greater than
(positive) or less than (negative) the first group (low motivation) on the
latent variables.
A latent variable mean difference value of 10.08 is indicated for the first
latent variable (City1), which indicates a statistically significant mean difference (i.e., high motivation group) had mean production rates greater
than the low motivation group in Los Angeles (City1).
A latent variable mean difference value of 10.18 is indicated for the second latent variable (City2), which indicates a statistically significant mean
difference (i.e., high motivation group) had mean production rates greater
than the low motivation group in Chicago (City2).
Overall, the high motivation groups outperformed the low motivation
groups in both cities. City1 and City2 correlated .90, indicating similar

Y102005.indb 467

3/22/10 3:27:46 PM

468

Answers to Selected Exercises

mean difference production rates. The latent variable mean differences
are divided by their standard error to yield a one-sample T value (i.e.,
10.08/.08 = 122.17, within rounding error).
Mean Vector of Independent Variables
City1
-------10.08
(0.08)
122.17

City2
-------10.18
(0.08)
128.16

Chapter 14
1. Second-Order Factor Analysis
The psychological research literature suggests that drug use and depression are leading indicators of suicide among teenagers. The following
LISREL–SIMPLIS program was run to test a second-order factor model.
Second Order Factor Analysis Exercise
Observed Variables: drug1 drug2 drug3 drug4 depress1
depress2 depress3 depress4
Sample Size 200
Correlation Matrix
1.000
0.628 1.000
0.623 0.646 1.000
0.542 0.656 0.626 1.000
0.496 0.557 0.579 0.640 1.000
0.374 0.392 0.425 0.451 0.590 1.000
0.406 0.439 0.446 0.444 0.668
.488 1.000
0.489 0.510 0.522 0.467 0.643
.591
.612 1.000
Means 1.879 1.696 1.797 2.198 2.043 1.029 1.947 2.024
Standard Deviations 1.379 1.314 1.288 1.388 1.405 1.269
1.435 1.423
Latent Variables: drugs depress suicide
Relationships
drug1 - drug4 = drugs
depress1 - depress4 = depress
drugs = Suicide
depress = Suicide
Set variance of drugs - Suicide to 1.0
Path diagram
End of problem

Y102005.indb 468

3/22/10 3:27:46 PM

469

Answers to Selected Exercises

The second-order factor model with standardized coefficients had an
acceptable fit (Chi-square = 30.85, df = 19, p = .042) and is diagrammed as:

0.74
drugs

0.81
0.79

0.98

1.00

0.82

drug1

0.45

drug2

0.34

drug3

0.35

drug4

0.38

depress1

0.25

depress2

0.53

depress3

0.44

depress4

0.39

Suicide

0.82

0.87
depress

0.69
0.75
0.78

The structure coefficients indicate that the first factors are strong indicators of the second factor (suicide). Drug use (R-squared = .96) was the
stronger indicator of suicide among teenagers.
Structural Equations
drugs = 0.98*Suicide, Errorvar. = 0.044, R² = 0.96
(0.17)
0.26
depress = 0.82*Suicide, Errorvar. = 0.33, R² = 0.67
(0.12)
(0.13)
6.96
2.51
NO T E :

Missing t-values and standard errors in SIMPLIS output.

Second-Order Factor Analysis—Suicide example.
Since the ETA variables (drugs and depress) are indicators of the corresponding KSI variable (suicide), LISREL by default fixes the loading of the

Y102005.indb 469

3/22/10 3:27:47 PM

470

Answers to Selected Exercises

first indicator to one. Then, after convergence the value of 1 is rescaled
using the estimated ETA1 variance. Although the corresponding standard
error estimate can be computed using the Delta method, LISREL does not
compute it. As a result, no standard error estimate and t value is written to
the output file. The LISREL 8 syntax program with a raw data file should
produce the standard errors and t-value.
3. MULTITRAIT–MULTIMETHOD MODELS
a. The LISREL–SIMPLIS program to analyze the three methods
(student, teacher, and peer) and three traits (behavior, motivate, and attitude) as a MTMM model using start values and
admissibility check off (increase iterations to achieve convergence) is:
MTMM Model Exercise
Observed Variables: X1 X2 X3 X4 X5 X6 X7 X8 X9
Correlation Matrix
1.0
.40 1.0
.31
.38 1.0
.35
.23
.16 1.0
.26
.22
.21
.62 1.0
.15
.11
.15
.49
.62 1.0
.43
.31
.24
.61
.48
.33 1.0
.40
.35
.19
.49
.45
.32
.74 1.0
.26
.20
.18
.43
.41
.33
.52
.47 1.0
Sample Size: 300
Latent Variables: behavior motivate attitude student
teacher peer
Relationships:
X1 = (.3)*behavior + (.5)*student
X2 = (.3)*motivate + (.5)*student
X3 = (.3)*attitude + (.5)*student
X4 = (.3)*behavior + (.5)*teacher
X5 = (.3)*motivate + (.5)*teacher
X6 = (.3)*attitude + (.5)*teacher
X7 = (.3)*behavior + (.5)*peer
X8 = (.3)*motivate + (.5)*peer
X9 = (.3)*attitude + (.5)*peer
Set variance of behavior - peer to 1.0
Set correlation of student and behavior to 0
Set correlation of student and motivate to 0
Set correlation of student and attitude to 0
Set correlation of teacher and behavior to 0
Set correlation of teacher and motivate to 0

Y102005.indb 470

3/22/10 3:27:47 PM

471

Answers to Selected Exercises

Set correlation of
Set correlation of
Set correlation of
Set correlation of
Options: AD = FF
Path diagram
End of problem

teacher and attitude
peer and behavior to
peer and motivate to
peer and attitude to

to 0
0
0
0

The MTMM model is displayed after dragging the three methods to the
left side of the diagram in the LISREL graph. The MTMM model had
acceptable fit indices (Chi-square = 10.85, df = 12, and p = .54).

X1
0.51
student

0.60
0.59

X2
X3

0.39

0.85

–0.06

–0.41
–0.65

0.04
–0.42

0.74
0.74
0.65
peer

0.62

X7
X8

motivate

–0.12

X4
0.69
teacher

0.48

–0.28

0.80

0.79

attitude

1.18

–0.18
–0.65
behavior

0.60
X9
Chi-Square = 10.85, df = 12, P-value = 0.54203, RMSEA = 0.000

The MTMM model results are displayed in Table C.1 to help the interpretation of trait and method effects. The assessment of Attitude regardless of which method was used had the higher error variance; Student
ratings (error = .64), Teacher ratings (error = .46), or Peer ratings (error =
.61), thus Attitude was the most difficult trait to assess, based on the three
methods used. The student and teacher rating methods were higher for
motivate (factor loading = .60 and factor loading = .85, respectively). The
peer rating method worked best with behavior, but was fairly similar
across all traits.

Y102005.indb 471

3/22/10 3:27:47 PM

472

Answers to Selected Exercises

Table C.1
MTMM Estimates of Three Methods on Three Traits (N = 300)
Traits
Behavior
Behavior
Motivate
Attitude

−.41

Behavior
Motivate
Attitude

–.41

Behavior
Motivate
Attitude

–.65

Motivate

Methods
Attitude

Student

–.06

.51
.60
.59

–.28

–.12
.04
–.65
–.18

Teacher

Peer

Error
.57
.56
.64

.68
.85
.74

.35
.27
.46
.65
.62
.60

.15
.19
.61

b. The LISREL–SIMPLIS program to run a Correlated Traits–
Correlated Uniqueness Model (CTCU) is:
Correlated Traits–Correlated Uniqueness Model Exercise
Observed Variables: X1 X2 X3 X4 X5 X6 X7 X8 X9
Correlation Matrix
1.0
.40 1.0
.31
.38 1.0
.35
.23
.16 1.0
.26
.22
.21
.62 1.0
.15
.11
.15
.49
.62 1.0
.43
.31
.24
.61
.48
.33 1.0
.40
.35
.19
.49
.45
.32
.74 1.0
.26
.20
.18
.43
.41
.33
.52
.47 1.0
Sample Size: 240
Latent Variables: behavior motivate attitude
Relationships:
X1 = behavior
X2 = motivate
X3 = attitude
X4 = behavior
X5 = motivate
X6 = attitude
X7 = behavior
X8 = motivate
X9 = attitude
Set variance of behavior - attitude to 1.0
Let error covariance of X1–X3 correlate

Y102005.indb 472

3/22/10 3:27:48 PM

473

Answers to Selected Exercises

Let error covariance of X4–X6 correlate
Let error covariance of X7–X9 correlate
Path diagram
End of problem

The CTCU model is diagrammed as:

0.76

0.23
0.19

0.85

0.28
0.92

behavior

X3
0.39

0.51
0.24
0.23

0.65

0.38
0.80

0.23

0.10
–0.00

0.37

0.92

X4
0.88
0.60
0.29

motivate

0.85

0.91

0.79

attitude
X8

–0.04
0.51

The results are presented in Table C.2. Findings indicated that all three
traits were statistically significantly correlated. More importantly, the peer
method was the best for assessing any of the three traits, as indicated
by the higher trait factor loadings and lower correlated uniqueness error
terms. Students are probably not rating themselves well and teachers
seemed a little better at rating student behavior and motivation than attitude. The data also had an acceptable fit to the CTCU model (c2 = 13.43, p =
.57, df = 15; RMSEA = .000; n = 300).

Y102005.indb 473

3/22/10 3:27:48 PM

474

Answers to Selected Exercises

Table C.2
Correlated Uniqueness Model with Correlated Traits and Errors
Method
Student

Teacher

Peer

Trait
Behavior
Motivate
Attitude
Behavior
Motivate
Attitude
Behavior
Motivate
Attitude

Trait correlations
Behavior
1.0
Motivate
.92
Attitude
.85

Factor
Loading

Uniqueness

.49
.39
.29
.70
.60
.45
.88
.79
.70

.76
.85
.92
.51
.65
.80
.23
.37
.51

.24
.15
.08
.49
.35
.20
.77
.63
.49

1.0
.91

Correlated Uniqueness of
Error Terms
1.0
.23
.19
1.0
.24
.23
1.0
.10
.00

1.0
.28

1.0

1.0
.38

1.0

1.0
–.04

1.0

Note: C2 = 13.43, p = .57, df = 15; RMSEA = .000; n = 300.

The LISREL Program was run again to estimate a correlated trait (CT)
only model with no correlated error terms. To accomplish this, you simply
delete the following command lines:
Let Error Covariance of Var1–Var3 Correlate
Let Error Covariance of Var4–Var6 Correlate
Let Error Covariance of Var7–Var9 Correlate

The results yielded a nonpositive definite matrix among the latent variables (i.e., correlations were greater than 1.0.) Also, the modification indices suggested adding the very error covariance you deleted. So, the CT
Model is rejected in favor of the CTCU Model.
behavior
motivate
attitude

behavior
1.00
1.07
0.95

motivate

attitude

1.00
1.10

1.00

W_A_R_N_I_N_G: is not positive definite

Y102005.indb 474

3/22/10 3:27:48 PM

Answers to Selected Exercises

475

The Modification Indices Suggest to Add an Error Covariance
Between
X2
X3
X3
X5
X6
X6
X7
X7
X8
X8
X8

and
X1
X1
X2
X4
X4
X5
X5
X6
X4
X6
X7

Decrease in Chi-Square
19.8
12.6
25.0
22.3
17.3
64.2
32.9
20.5
23.6
11.2
49.7

New Estimate
0.22
0.17
0.26
0.18
0.16
0.34
–0.22
–0.16
–0.18
–0.14
0.27

Although the MTMM model achieved an acceptable model fit, the findings were mixed as to which method worked best with the three traits
(behavior, motivate, and attitude). The CTCU model in contrast more
clearly indicated that peers did a better job of rating the traits. Students
tend to know other students more on these traits both in and outside the
classroom, thus providing a theoretical argument for the findings.

Chapter 15
1. Multiple Indicator and Multiple Cause Model
The following LISREL–SIMPLIS program would be created and run to
determine the parameter estimates and model fit.
MIMIC Model of Job Satisfaction
Observed Variables peer self income shift age
Sample Size 530
Correlation Matrix
1.00
.42 1.00
.24
.35 1.00
.13
.37
.25 1.00
.33
.51
.66
.20 1.00
Latent Variable satisfac
Relationships
peer = satisfac
self = satisfac
satisfac = income shift age
Path diagram
End of problem

Y102005.indb 475

3/22/10 3:27:49 PM

476

Answers to Selected Exercises

Initial MIMIC Model Results
The MIMIC model results indicated an adequate fit with chi-square = 6.81,
df = 2, and p = .033. The measurement equations indicated that job satisfaction (satisfac) was adequately defined with self ratings being a better
indicator of job satisfaction than peer ratings.
Measurement Equations
peer = 0.48*satisfac, Errorvar. = 0.77, R² = 0.23
(0.053)
14.49
self = 0.87*satisfac, Errorvar. = 0.25 , R² = 0.75
(0.11)
(0.078)
8.10 3.16
The structural equation indicated that 45% of job satisfaction was predicted by knowledge of income, what shift a person worked, and their
age. However, the coefficient for income was not statistically significant
(T = −.59). Consequently the model should be modified by dropping this
variable and re-running the analysis.
Structural Equations
satisfac = – 0.032*income + 0.31*shift + 0.56*age, Errorvar.= 0.55, R² = 0.45
(0.054)
(0.054)
(0.082)
(0.11)
–0.59 5.71 6.77
5.14
MIMIC Modification
The MIMIC model modification resulted in little improvement with chisquare = 6.11, df = 1, and p = .01. The measurement equations were not very
different. Other measures would help to define the latent variable, job satisfaction. The structural equation resulted in the same R-squared value, which
indicates that income did not add to the prediction of job satisfaction. A parsimonious model was therefore achieved, but the 55% unexplained variance
implies that other variables could be discovered to increase prediction.
Measurement Equations
peer = 0.49*satisfac, Errorvar. = 0.76, R² = 0.24
(0.053)
14.48
self = 0.87*satisfac, Errorvar. = 0.25, R² = 0.75
(0.11)
(0.078)
8.12
3.21

Y102005.indb 476

3/22/10 3:27:49 PM

Answers to Selected Exercises

477

Structural Equations
satisfac = 0.31*shift + 0.54*age, Errorvar.= 0.55 , R² = 0.45
(0.053)
(0.073)
(0.11)
5.72
7.39
5.14

3. Multilevel Model
The multilevel analysis of data in the PRELIS system file, income.psf, was
used with the pull down multilevel menu to create and run 3 different
PRELIS programs. Results are summarized in a table with the intra-class
correlation (hand computed) for comparative purposes.
Model 1 is the baseline model (constant), followed by the added effects
of gender, and the added effects of marital status (marital). The 3 different
PRELIS programs should look as follows:
Model 1 (intercept only)
OPTIONS OLS=YES CONVERGE=0.001000 MAXITER=10 OUTPUT=STANDARD ;
TITLE=income decomposition;
SY=’C:\LISREL 8.8 Student Examples\MLEVELEX\INCOME.PSF’;
ID3=region;
ID2=state;
RESPONSE=income;
FIXED=constant;
RANDOM2=constant;
RANDOM3=constant;
Model 2 (intercept + gender)
OPTIONS OLS=YES CONVERGE=0.001000 MAXITER=10 OUTPUT=STANDARD ;
TITLE=income decomposition;
SY=’C:\LISREL 8.8 Student Examples\MLEVELEX\INCOME.PSF’;
ID3=region;
ID2=state;
RESPONSE=income;
FIXED=constant gender;
RANDOM2=constant;
RANDOM3=constant;
Model 3 (intercept + gender + marital)
OPTIONS OLS=YES CONVERGE=0.001000 MAXITER=10 OUTPUT=STANDARD ;
TITLE=income decomposition;
SY=’C:\LISREL 8.8 Student Examples\MLEVELEX\INCOME.PSF’;
ID3=region;

Y102005.indb 477

3/22/10 3:27:49 PM

478

Answers to Selected Exercises

ID2=state;
RESPONSE=income;
FIXED=constant gender marital;
RANDOM2=constant;
RANDOM3=constant;

The PRELIS program results for the three analyses are summarized in
Table C.1. The baseline model (intercept only) provides the initial breakdown of level 3 and level 2 error variance. The multilevel model for the
added effect of gender is run next. The chi-square difference between Model
1 and Model 2 yields chi-square = 5.40, which is statistically significant at
the .05 level of significance. Gender, therefore, does help explain variability
in income. Finally, marital is added to the multilevel model, which yields a
chi-square difference between Model 2 and Model 3 of chi-square = 1.18.
The chi-square difference value is not statistically significant; therefore,
marital status does not add any additional significant explanation of variability in income.

Table C.3
Summary Results for Multilevel Analysis of Income
Multilevel Model
Fixed Factors

Model 1
Constant

Model 2 Constant +
Gender

Model 3 Constant +
Gender + Marital

Intercept Only(B0)
Gender (B1)
Marital (B2)

10.096 (.099)

10.37 (.15)
–0.42 (.16)

10.24 (.19)
–0.43 (.16)
.19 (.17)

.37

.31

.30

.02

.05

.06

.051 (5%)
11144.29
3

.138 (14%)
11138.89
4
5.40

.166 (17%)
11137.71
5
1.18

Level 2 error
variance (eij)
Level 3 error
variance (uij)
ICC
Deviance (–2LL)
Df
χ 2 Difference (df = 1)

Note: χ2 = 3.84, df = 1, p = .05.
Note: ICC1 =

Y102005.indb 478

Φ3
Tau − Hat(Level − 3)
.02
=
=
= .051
Φ 3 + Φ 2 Tau − Hat(Level − 3) + Tau − Hat(Level − 2) .02 + .37

3/22/10 3:27:49 PM

Answers to Selected Exercises

479

Chapter 16
1. Interaction Model
An organizational psychologist was investigating whether work tension
and collegiality were predictors of job satisfaction. However, research indicated that work tension and collegiality interact, so a SEM Interaction Model
was hypothesized and tested. [Note: You need to use a raw data file so that
values for latent variables can be added.]
First open the PRELIS system file, jobs.psf, to view the 9 observed variables.

Second, create the LISREL–SIMPLIS program to create and save the
latent variables in the PRELIS system file, jobs.psf:
Computing Latent Variable Scores
Observed Variables v1-v9
Raw Data from File jobs.psf
Latent Variables : job work colleg
Relationships:
v1=1*job
v2-v3= job
v4=1*work
v5-v6=work
v7=1*colleg

Y102005.indb 479

3/22/10 3:27:50 PM

480

Answers to Selected Exercises

v8-v9=colleg
PSFfile jobs.psf
End of problem
N ote : Remember to close the PRELIS system file, jobs.psf, and then open it
again to see that the latent variables have been added.

Third, create the latent interaction variable by using the
TRANSFORMATION, then COMPUTE on the pull down menu. Select
ADD, enter name for new variable (interact), then drag variable names to
the Compute window (interact=work*colleg).

Y102005.indb 480

3/22/10 3:27:50 PM

Answers to Selected Exercises

481

Click OK, and the latent interaction variable, interact, will automatically
be added to the PRELIS system file, jobs.psf.

Finally, create and run a LISREL–SIMPLIS program to analyze the
Interaction Model.
Latent Interaction Variable Model - No Intercept Term
Observed Variables: v1-v9 job work colleg interact
Raw Data from File jobs.psf
Sample Size = 200
Relationships:
job = work colleg interact
Path diagram
End of problem

The structural equation indicates that no interaction effect is present
between work tension and collegiality. Rather, work tension and collegiality
are predictors of job satisfaction as direct linear effects.
Structural Equations
job = 0.98*work – 0.18*colleg + 0.036*interact, Errorvar.= 0.22 , R² = 0.80
(0.065) (0.079) (0.038) (0.022)
15.16 –2.29 0.96
9.90
The latent interaction variable should be dropped and the LISREL–SIMPLIS
program run again. The R-squared value does not change indicating that the
interaction effect did not contribute to the prediction of job satisfaction.
Structural Equations
job = 0.97*work – 0.17*colleg, Errorvar. = 0.22 , R² = 0.80
(0.064) (0.078) (0.022)
15.20 –2.17
9.92

Y102005.indb 481

3/22/10 3:27:51 PM

482

Answers to Selected Exercises

3. MONTE CARLO Methods
The SPSS program would input the population matrix values as follows:
MATRIX.
compute popr =
{1, .50, .30, .90;
.50, 1, .70, .50;
.30, .70, 1, .50;
.90, .50, .50, 1}.
Print popr.
compute pi = 3.14159.
compute rown = nrow(popr).
compute n = 10000.
compute corr = sqrt(–2*ln(uniform(n,rown)))&*cos((2*pi)*uniform(n,rown)).
compute corr=corr*chol(popr).
save corr /outfile = pop.sav.
END MATRIX.
The SPSS output would look like this:
Run MATRIX procedure:
POPR
1.000000000 .500000000 .300000000 .900000000
.500000000 1.000000000 .700000000 .500000000
.300000000 .700000000 1.000000000 .500000000
.900000000 .500000000 .500000000 1.000000000
------ END MATRIX ----You would now open the pop.sav file which would look like the following
(Note: Our pop.sav file was in c:\program files\spssinc\spss16 folder).

Y102005.indb 482

3/22/10 3:27:51 PM

483

Answers to Selected Exercises

The SPSS correlation procedure was selected and ran:

The output from the SPSS correlation procedure yielded population correlation values in Table C.4 similar to what we specified in the SPSS MATRIX
program.
Table C.4
Correlations
COL1
COL1

COL2

COL3

COL4

Pearson Correlation
Sig. (1-tailed)
N
Pearson Correlation
Sig. (1-tailed)
N
Pearson Correlation
Sig. (1-tailed)
N
Pearson Correlation
Sig. (1-tailed)
N

COL2

1.000
10000.000
.489**
.000
10000
.287**
.000
10000
.899**
.000
10000

.489**
.000
10000
1.000
10000.000
.696**
.000
10000
.488**
.000
10000

COL3
.287**
.000
10000
.696**
.000
10000
1.000
10000.000
.491**
.000
10000

COL4
.899**
.000
10000
.488**
.000
10000
.491**
.000
10000
1.000
10000.000

Correlation is significant at the 0.01 level (1-tailed).

Y102005.indb 483

3/22/10 3:27:51 PM

484

Answers to Selected Exercises

Chapter 17
1. The diagrammed structural equation model is shown as
Figure 17.4.
err_1

Income

err_2

FAED

err_3

MOED

err_4

Personal

err_5

School

err_6

Admit

err_13

Student

Family

Cognate

err_7

Self

err_8

Motivate

err_9

Occup

err_10

Educ

err_11

Aspire

Encourag

err_12

Figure 17.4
Student characteristic model.

The measurement equations are as follows:
For the X variables using variable names
family income = function of Family Background + error
father’s education = function of Family Background + error
mother’s education = function of Family Background + error
personal encouragement = function of Encouragement + error
institutional characteristics = function of Encouragement + error
admission status = function of Encouragement + error.
The measurement equations for the Xs are
X1 = 1.0 x1 + d 1
X2 = lx21 x1 + d 2
X3 = lx31 x1 + d 3
X4 = 1.0 x2 + d 4

Y102005.indb 484

3/22/10 3:27:52 PM

485

Answers to Selected Exercises

X5 = lx52 x2 + d 5
X6 = lx62 x2 + d 6.
The matrix equations for the Xs are
 X1   1
  
 X 2   λ x21
 X 3   λ x31
 =
X4   0
X   0
 5 
 X 6   0

δ 1 
0 

 
0 
δ
ξ1   2 
0    δ 3 
 +  
1    δ 4 
ξ 
λ x52   1  δ 5 

 
λ x62 
δ 6 

For the Y variables using variable names:
cognitive abilities = function of Student Characteristics + error
interpersonal skills = function of Student Characteristics + error
motivation = function of Student Characteristics + error
occupational aspirations = function of Aspiration + error
educational aspirations = function of Aspiration + error.

The measurement equations for the Ys are
Y1 = 1.0 h1 + e1
Y2 = ly21 h1 + e 2
Y3 = ly31 h1 + e 3
Y4 = 1.0 h2 + e 4
Y5 = ly52 h2 + e 5.
The matrix equations for the Ys are
 Y1   1
  
Y2   λ y 21
Y3   λ y 31
 =
Y4   0
Y   0
 5 
Y6   0

0 

0 
0 

1 
λ y 52 

λ y62 

η1 
 
+
η1 
 

 ε1 
 
ε 2 
ε 3 
 
ε 4 
ε 
 5
 ε 6 

The structural equations using variable names are
Students’ Characteristics = Family Background + Encouragement + Aspirations + error
Aspirations = Family Background + Encouragement + error.

Y102005.indb 485

3/22/10 3:27:53 PM

486

Answers to Selected Exercises

The structural equations are written as

η1 = β12η2 + γ 11ξ1 + γ 12ξ2 + ζ 1
η2

= γ 21ξ1 + γ 22ξ2 + ζ 2 .

In matrix form the structural equations are
 η1   0
 =
η2   0

β12 

0 

 η1  γ 11
 +
η2  γ 21

γ 12 

γ 22 

 ξ1   ξ1 
 + 
ξ2  ξ2 

The matrix of the structural coefficients for the endogenous variables is
0
B=
0

β12 

0 

The matrix of the structural coefficients for the exogenous variables is
γ
Γ =  11
γ 21

γ 12 

γ 22 

The matrix of the factor loadings for the endogenous variables is
 1

 λ y 21
Λ y =  λ y 31

 0
 0


0 

0 
0 

1 
λ y 52 

The matrix of the factor loadings for the exogenous variables is
 1

 λ x21
 λ x31
Λx =  0

 0


 0

0 

0 
0 

1 
λ x52 


λ x62 

The covariance matrix for the exogenous latent variables is
φ
Φ =  11
φ21

Y102005.indb 486



φ22 

3/22/10 3:27:54 PM

487

Answers to Selected Exercises

The covariance matrix for the equation errors is
ψ
ψ =  11
 0



ψ 22 

The covariance matrix for the measurement errors of the indicators of the
exogenous latent variables is
θδ 11

 0
 0
Θδ = 
 0
 0

 0

θδ 22
0
0
0
0

θδ 33
0
0
0

θδ 44
0
0

θδ 55
0









θδ 66 

The covariance matrix for the measurement errors of the indicators of the
endogenous latent variables is
θε 11

 0
Θε =  0

 0
 0


θε 22
0
0
0

θε 33
0
0

θε 44
0







θε 55 

The structural equation model can be interpreted from the direct and
indirect effects to yield the total effects for the model. The direct effects
for Aspirations are Family Background (g 21) and Encouragement (g 22).
The direct effects for Students’ Characteristics are Family Background
(g 11), Encouragement (g 12), and Aspirations (b12). The indirect effects for
Students’ Characteristics is Family Background through Aspirations (g 21 b12).
Thus, the total effects are as follows:
Family Background -> Aspirations = g 21
Encouragement -> Aspirations = g 22
Family Background -> Students’ Characteristics = g 11 + (g 21) (b12)
Encouragement -> Students’ Characteristics = g 12
Aspirations -> Students’ Characteristics = b12.

Y102005.indb 487

3/22/10 3:27:55 PM

Author Index
A
Aalberts, C., 286, 292
Acock, A. C., 342, 370
Adamson, G., 290
Aiken, L. S., 341, 369
Akaike, H., 90, 120
Algina, J., 33, 52, 277, 291, 341, 369, 370
Alstine, J. V., 91, 122, 193
American Psychological Association,
209, 220
Anderson, C., 39, 51
Anderson, J. C., 42, 51, 92, 114, 120,
191, 193
Anderson, K. G., 324
Anderson, N., 267
Anderson, N. H., 18, 31
Anderson, R. E., 76, 121
Anderson, T. W., 5, 11
Andrews, J. A., 291
Arbuckle, J. L., 38, 51, 230, 243, 250, 269
Arvey, R., 263, 269, 341, 369
Ary, D., 291
Aurelio, J. F., 21, 31, 38, 52
Austin, J. T., 210, 221
B
Bagley, M. N., 325
Bailey, L. R., 368
Baker, R. L., 48, 53
Baldwin, B., 66, 69, 75, 120
Balla, J. R., 91, 122
Bandalos, D. L., 230, 243, 365, 369
Bang, J. W., 346, 369
Bashaw, W. L., 125, 139
Beale, E. M. L., 38, 51
Bennett, N., 91, 122, 193
Benson, J., 230, 231, 243
Bentler, P. M., 42, 51, 52, 61, 66, 69, 70,
75, 77, 88, 89, 114, 119, 120, 121,
122, 123, 229, 243
Beukema, J. J., 290

Black, W. C., 76, 121
Blalock, H. M., 5
Bohrnstedt, G. W., 136, 138, 140
Bolding, J. T., Jr., 126, 140
Bollen, K. A., 58, 69, 77, 87, 99, 113, 120,
144, 161, 229, 240, 243, 283,
286, 291, 337, 340, 341, 365,
369, 371
Bonett, D. G., 75, 88, 120
Boomsma, A., 42, 47, 51, 210, 220
Breckler, S. J., 209, 220
Brett, J. M., 90, 121, 191, 193
Broaddus, M. R., 325
Brown, R., 130, 141
Browne, M. W., 47, 51, 61, 67, 69, 70, 76,
106–107, 108, 119, 120, 121,
122, 230, 231, 243
Bryan, A., 325
Buchner, A., 95, 121
Bullock, H. E., 48, 51
Bunting, B. P., 290
Byrne, B. M., 116, 120, 286, 291, 341
C
Cai, L., 108, 122
Campbell, D. T., 277, 291
Carter, T. M., 136, 138, 140
Chan, D. W., 290
Chatterjee, S., 130, 140
Chen, F., 365, 371
Cheung, D., 290
Cheung, G. W., 116, 120
Chou, C., 42, 51, 66, 70
Chow, S. M., 290
Cleary, T. A., 136, 140
Clelland, D. A., 144, 161
Cliff, N., 93, 120
Cochran, W. G., 136, 137, 140, 163, 177
Cohen, J., 42, 46, 51, 113, 120, 126,
128, 140
Cohen, P., 42, 46, 51, 126, 140

489

Y102005.indb 489

3/22/10 3:27:55 PM

490

Cole, D. A., 209, 221, 229, 243, 263, 269,
277, 291, 341, 369
Collins, L. M., 48, 51
Comrey, A. L., 164, 177
Conger, R. D., 290
Conner, B. T., 267
Cooley, W. W., 55, 70
Corneal, S. E., 291
Costello, A. B., 42, 51, 116, 164, 177
Costner, H. L., 66, 70
Crocker, L., 33, 52
Crombie, G., 341, 369
Cudeck, R., 6, 11, 67, 70, 76, 114, 120,
230, 231, 243
Cumming, G., 128, 140
Curran, P. J., 365, 371
D
Darlington, R. B., 135, 140
Davey, A., 21, 31, 38, 52, 114, 121
Dekker, R., 290
Delucchi, M., 4, 11
Ding, L., 42, 52
Dorans, N. J., 62, 71
Draper, N. R., 125, 140
Drasgow, F., 62, 71
Drezner, Z., 66, 71, 74, 122
Duncan, O. D., 5, 151, 152, 161
Duncan, S. C., 342, 368, 369, 370
Duncan, T. E., 342, 368, 369, 370
Dupont, W. D., 368
du Toit, M., 324, 325, 337, 341, 369
du Toit, S., 6, 11, 324, 325, 337,
341, 369
Duval, R. D., 240, 244, 340, 370
E
Edwards, A. L., 125, 140
Ehman, L. H., 21, 31, 38, 53
El-Zahhar, N., 231, 243
Enders, C. K., 21, 31, 38, 52, 365, 369
Epstein, D., 229, 244, 341, 370
Erdfelder, E., 95, 121
Etezadi-Amoli, J., 341, 369
Everson, H. T., 325

Y102005.indb 490

Author Index

F
Fabrigar, L. R., 57, 71
Fan, X., 62, 70, 234, 243, 365, 368, 369, 370
Faul, F., 95, 121
Faulbaum, F., 229, 243
Felsovalyi, A., 365, 369
Ferguson, G. A., 33, 36, 52
Fielding, D., 368
Finch, S., 128, 140
Findley, W. G., 125, 139
Fiske, D. W., 277, 291
Fouladi, T., 128, 141
Fryer, R. J., 290
Fuller, W. A., 136, 137, 140, 163, 177
Furr, M. R., 282, 291
G
Gallini, J. K., 66, 70
Galton, Sir Francis, 33, 34
Ge, X., 290
Geary, D. C., 267
Gerbing, D. W., 42, 51, 61, 70, 92, 114,
116, 120, 121, 191, 193
Ghisletta, P., 368
Glymour, C. R., 66, 70, 72
Goldberg, L., 5, 11
Goldberger, A. S., 151
Gonzalez, R., 94, 121
Gorsuch, R. L., 77, 121, 164, 177
Grayson, D., 282, 286, 287, 291
Griffin, D., 94, 121
H
Hagglund, G., 61, 70
Hahn, L. W., 368
Hair, J. F., Jr., 76, 121
Hamilton, J. G., 61, 70, 116, 121
Hancock, G., 267
Harlow, L. L., 42, 48, 51, 52
Harmon, H. H., 77, 121, 164, 177
Harwell, M., 21, 31, 38, 53
Hau, K.-T., 91, 122
Hayashi, N., 267
Hayduk, L. A., 229, 243, 341, 370, 373,
375, 423

3/22/10 3:27:55 PM

491

Author Index

Heck, R. H., 307, 325
Henly, S. J., 114, 120
Herbing, J. R., 66, 70
Hershberger, S. L., 6, 11, 57, 70, 291
Hidiroglou, M. A., 136, 140
Higgins, L. F., 341, 370
Hinkle, D. E., 46, 49, 52, 125, 140
Ho, K., 39, 52
Hoelter, J. W., 41, 52, 99, 121
Holahan, P. J., 90, 123
Holbert, R. L., 368
Holzinger, K. J., 77, 121, 164, 177, 212, 220
Hops, H., 291, 342, 370
Horn, J. L., 48, 51
Houston, S. R., 126, 140
Howe, W. G., 5, 11
Hox, J., 307, 325
Hoyle, R. H., 209, 221
Hu, L., 42, 52, 77, 119, 121
Huber, P. J., 39, 52
Huberty, C. J., 127, 129, 135, 140
Huelsman, T. J., 282, 291

Keesling, W., 6, 12, 59, 70
Kelly, F. J., 125, 141
Kelly, K., 66, 70
Kenny, D. A., 91, 121, 341, 370, 411, 423
Kerlinger, F. N., 127, 140
Kirby, J., 365, 371
Kopriva, R., 130, 141
Kroonenberg, P. M., 67, 70, 290
L

James, L. R., 90, 91, 121, 122, 191, 193
Jennrich, R. I., 47, 52
Jolliffe, I. T., 290
Jöreskog, K. G., 5, 6, 11, 12, 19, 31, 49, 52,
58, 59, 65, 70, 77, 85, 116, 121,
133, 136, 138, 140, 141, 144, 161,
164, 177, 183, 191, 193, 199, 208,
224, 229, 234, 237, 243, 259,
269, 293, 298, 301, 304, 324,
325, 337, 340, 341, 370, 371,
373, 375, 383, 405, 423
Judd, C. M., 341, 370, 411, 423
Jurs, S. G., 46, 49, 52, 125, 140

Lang, A.-G., 95, 121
Lawley, D. N., 5, 12, 47, 52
Leamer, E. E., 64, 70
Lee, H. B., 164, 177
Lee, S., 57, 70
Lei, M., 62, 70
Leroy, A. M., 39, 53
Levin, J. R., 136, 141
Lewis, C., 67, 70, 88, 123
Li, F., 342, 370
Liang, J., 99, 120
Lievens, F., 267, 291
Lim, B., 290
Lind, J. M., 89, 123
Lind, S., 91, 122, 193
Linn, R. L., 136, 141
Liou, S.-M., 21, 31, 38, 53
Little, R. J., 38, 51, 52
Loehlin, J. C., 86, 88, 121, 137, 140
Loken, E., 325
Lomax, R. G., 6, 12, 37, 46, 47, 49, 52, 61,
62, 66, 69, 70, 71, 114, 121, 125,
140, 198, 203, 208, 229, 244,
250, 269, 277, 291
Long, B., 267
Long, J. S., 77, 341, 365, 370, 371
Longshore, D., 267
Lubke, G. H., 325
Ludtke, O., 325
Lunneborg, C. E., 93, 121, 240, 244,
340, 370
Lyons, M., 136, 140

Kano, Y., 42, 52, 119, 121
Kaplan, D., 49, 53, 66, 70, 113, 121
Keenan, S. C., 365, 369

MacCallum, R. C., 57, 66, 71, 106–107,
108, 121, 122, 210, 221
Mackenzie, S. B., 341, 370

I
Igarashi, Y., 267
Iniesta-Bonillo, M. A., 324
J

Y102005.indb 491

3/22/10 3:27:56 PM

492

Mallinckrodt, B., 267
Mallows, C. L., 135, 141
Mandeville, G. K., 66, 70
Marchant, G. J., 341, 371
Marcoulides, G. A., 66, 71, 74, 122, 245,
271, 291, 292, 341, 371, 410,
414, 423
Marsh, H. W., 89, 91, 113, 115, 117, 122,
282, 286, 287, 291
Maxwell, A. E., 52
Maxwell, S. E., 47, 209, 221, 229, 243,
263, 269, 277, 291, 341, 369
McArdle, J. J., 229, 244, 290, 341, 368, 370
McCall, C. H., Jr., 41, 52
McCarthy, D. M., 324
McCoach, D. B., 91, 121
McDonald, J. A., 144, 161
McDonald, R. P., 89, 91, 115, 117, 122,
210, 221, 341, 369, 370
McKnight, K. M., 21, 31, 38, 52
McKnight, P. E., 21, 31, 38, 52
McNeil, J. T., 125, 141
McNeil, K. A., 125, 141
Meek, C., 66, 72
Millar, R., 368
Miller, L. M., 135, 141
Miller, M., 291
Millsap, R. E., 115–116, 122, 191,
193, 325
Mokhtarian, P. L., 325
Molenaar, P. C. M., 291
Mooijaart, A., 290
Mooney, C. Z., 240, 244, 340, 365, 370
Moore, J. H., 368
Morf, C. C., 267
Moulder, B. C., 341, 369, 370
Moulin-Julian, M., 231, 243
Mulaik, S. A., 48, 51, 90, 91, 114,
115–116, 121, 122, 191, 193
Mulhall, P. K., 290
Muthén, B., 49, 52, 53, 61–62, 71, 113,
114, 122, 229, 325, 369
Muthén, L., 113, 114, 122, 369
N
Naugher, J. R., 39, 52
Nemanick, R. C., Jr., 282, 291

Y102005.indb 492

Author Index

Nesselroade, J. R., 209, 290
Newman, I., 125, 141, 341, 371
Niggli, A., 325
O
Oakman, J. M., 291
Olsson, U., 62, 71
Osborne, J., 42, 51, 116, 120, 164, 177
P
Pajares, F., 291
Panter, A. T., 209, 221
Parkerson, J. A., 5, 12
Parl, F. F., 368
Patterson, G. R., 290
Paxton, P., 365, 371
Pearl, J., 48, 53, 144, 161
Pearson, E. S., 4, 12
Pearson, K., 4, 33, 34, 53
Pedhazur, E. J., 44, 46, 53, 126, 127, 138,
140, 141, 159, 160, 161
Penev, S., 57, 71
Peng, C.-Y.J., 21, 31, 38, 53
Ping, R. A., Jr., 341, 371
Ployhart, R. E., 290
Poon, W. Y., 267
Q
Quilty, L. C., 291
R
Rand, D., 290
Raykov, T., 57, 60, 71, 209, 221
Rensvold, R. B., 116, 120
Resta, P. E., 48, 53
Ridenour, T., 341, 371
Rigdon, E., 328, 371
Ringo Ho, M., 210, 221
Riski, E., 291
Ritchie, M. D., 368
Roche, R. A., 291
Rock, D. A., 136, 141
Roodi, N., 368
Rousseeuw, P. J., 39, 53

3/22/10 3:27:56 PM

493

Author Index

Rubin, H., 5, 11, 38, 52
Russell, D. W., 267
S
Salas, E., 263, 269, 341, 369
Sanchez-Perez, M., 324
Saris, W. E., 66, 71, 94, 113, 122,
286, 292
Sasaki, M. S., 340, 371
Satorra, A., 66, 71, 94, 113, 114, 119, 122
Savla, J., 21, 31, 38, 52, 114, 121
Scheines, R., 66, 70
Schiller, D. P., 12
Schlackman, J., 267
Schlieve, P. L., 346, 369
Schmelkin, L. P., 138, 141
Schmiege, S. J., 325
Schnyder, I., 325
Schumacker, R. E., 39, 66, 71, 74, 122,
135, 136, 137, 141, 245, 271,
291, 292, 328, 341, 346, 368,
369, 371, 410, 414, 423
Schwarzer, C., 231, 243
Seipp, B., 231, 243
Shapiro, A., 61, 71
Sheather, R. G., 39, 53
Shenzad, S., 324
Shevlin, M., 368
Shifren, K., 290
Shumow, L., 6, 12
Sidani, S., 21, 31, 38, 52
Silvia, E. S. M., 66, 71
Sivo, S., 365, 369
Skrondal, A., 365, 371
Smith, G. T., 324
Smith, H., 125, 140
Smith, K. W., 340, 371
Smith, Q. W., 135, 141
Smithson, M., 128, 141
Soper, D. S., 93, 122, 128, 141
Sörbom, D., 6, 11, 31, 49, 52, 58, 59, 65,
66, 70, 71, 72, 77, 85, 121, 133,
138, 140, 144, 161, 164, 177, 183,
191, 193, 199, 208, 224, 229, 234,
237, 243, 259, 269, 293, 298, 301,
304, 324, 325, 337, 340, 370, 373,
375, 383, 405, 423

Y102005.indb 493

Spearman, C., 5, 33, 34, 53
Specht, D. A., 158, 161
Spirtes, P., 66, 70, 72
Spreng, R. A., 341, 370
Staudte, R. G., 39, 53
Steiger, J. H., 60, 72, 76, 89, 122, 123,
128, 141
Stein, J. A., 267
Stephenson, M. T., 368
Stevens, S. S., 18, 31, 34, 35, 53
Stilwell, C. D., 91, 122, 193
Stine, R., 240, 244, 340, 371
Stoolmiller, M., 341, 371
Stronkhorst, L. H., 66, 71
Subkoviak, M. J., 136, 141
Suda, K., 267
Sugawara, H. M., 106–107, 121
Sullins, W. L., 425
Sunita, M. S., 116, 120
Sutcliffe, J. P., 136, 141
Swineford, F. A., 77, 121, 164, 177,
212, 220
T
Takane, Y., 33, 36, 52
Tang, F. C., 267
Tankard, J. W., Jr., 33, 34, 53
Tatham, R. L., 76, 121
Thayer, D. T., 47, 52
Thomas, S. L., 307, 325
Thompson, B., 135, 141, 210, 221
Thomson, W. A., 135, 141
Thurstone, L. L., 5
Tildesley, E. A., 291
Tippets, E., 66, 72
Tomer, A., 209, 221
Torres, S., 368
Tracz, S. M., 48, 53, 130, 141
Trautwein, U., 325
Tschanz, B. T., 267
Tucker, L. R., 88, 123
Turner, C. W., 267
U
Uchino, B. N., 57, 71
Unrau, N., 267

3/22/10 3:27:56 PM

494

V
van Dam, M., 290
Van Keer, E., 291
van Thillo, M., 6
van Uzendoorn, M. H., 290
Velicer, W. F., 42, 52

Author Index

Wold, H., 5
Wolfle, L. M., 147, 156, 158, 161
Wong, S. K., 341, 371
Wood, P. K., 66, 72
Wothke, W., 38, 40, 53, 230, 243, 250,
269, 282, 286, 287, 292
Wright, S., 5, 12, 143, 158, 161
Wu, E., 229, 243

W
Walberg, H. J., 5, 12
Wald, A., 59, 72
Wang, L., 62, 70
Watkins, D., 116, 120
Wegener, D. T., 57, 71
Wei, M. F., 267
Werts, C. E., 136, 141
West, S. G., 341, 369
Whitworth, R. H., 267
Widaman, K. F., 60, 71, 286, 292
Wiersma, W., 46, 49, 52, 125, 140
Wiley, D., 6, 12, 59, 72
Williams, L. J., 90, 123

Y102005.indb 494

Y
Yamashina, M., 267
Yang, F., 341, 370, 423
Yang-Wallentin, F., 340, 342, 370, 371
Yilmaz, M., 130, 140
Yuan, K.-H., 123
Z
Zakalik, R. A., 267
Zuccaro, C., 135, 141
Zuur, A. F., 290

3/22/10 3:27:56 PM

Subject Index
A

Addition of matrices, 426
Adjoint matrix, 431
Adjusted goodness-of-fit index
(AGFI), 87, 89, 205
Adjusted R 2, 127–128, 133–134
Akaike’s Information Criterion
(AIC), 42, 76–77, 89–91,
230
Alternative models, 73, 75, 88, 211
AMOS, 8
ANOVA repeated measure designs,
341–345
Answers to exercises, 449–487
Arcsine transformation, 36
Areas under the Normal Curve
(table), 440
ARIMA, 274
Asymptotic covariance matrix, 28–29,
35, 118
Asymptotic distribution-free (ADF)
estimators, 60, 61, 63
Asymptotic variance-covariance
matrix, 29, 62

California Achievement Test, 184–185
Categorical variable interaction
model, 328–331, 341
Categorical variable methodology
(CVM), 62
Categorical variables, 19, 29
mixture models, 298–307
Causal assumptions, 48–49
Causal modeling, 143
Cause-effect relationships, 48–49,
143
Chi-square, 85–86
badness-of-fit statistic, 86, 154,
203
confirmatory factor models, 172
critical, 99
distribution for given probability
(table), 443–444
estimation methods, 86
LISREL computation, 118–119
LISREL-SIMPLIS multiple sample
analysis, 225
model-fit criteria, 74, 75
path model-fit index, 158
reporting, 91
residual values and, 75
sample size and, 41, 86, 99–100,
211
Satorra-Bentler scaled robust
statistic, 62, 119, 305–306
Chi-square difference test, 116
categorical-variable interaction
model, 330
multiple group path model
analyses, 258
Cholesky decomposition, 352,
355–358
Class data, 29, 298
Comparative fit index (CFI), 42, 76,
89, 116
Confidence intervals (CIs), 128

B
Badness-of-fit statistic, 86, 154,
203, 230
Beta (β) matrix, 374–375, 376, 379,
380
Binary response variables, 61–62
Biserial correlations, 35
Bivariate correlations, 42
Bootstrap, 219, 234–240, 337
latent variable interaction model,
340
PRELIS and LISREL program
syntax, 237–240
PRELIS GUI, 234–237
Bootstrap estimator, 234
Browne-Cudeck criterion (BCC), 230

495

Y102005.indb 495

3/22/10 3:27:56 PM

496

Confirmatory factor analysis (CFA), 5,
163–164
exploratory factor analysis versus,
164
four-step approach and, 115
Confirmatory factor models, 163–164,
184, See also Measurement
models
example, 164–166
LISREL-SIMPLIS program, 174–176
measurement error, 165–166
misspecified model, 169–174
model estimation and, 169–170
model identification and, 167–169
model modification, 173–175
model specification and, 166–167
model testing, 170–173
multiple samples, 224
parameter estimation, 168–170
population model specification,
360, 363–364
variables in, 4
Confirmatory models, 73
Constrained parameters, 57, 380
Construct validity coefficient, 277
Content validity, 182
Continuous variables, 19, 29, 298
interaction models, 330, 339
mixture models, 298–307
nonlinear relations, 327–328
Convergent validity, 114, 182, 191,
277–278
Correction for attenuation, 39, 50, 137
Correlated error covariance, 303–304
Correlated measurement error, 190,
197, 274, 345
Correlated trait-correlated uniqueness
(CTCU) model, 282–286
Correlated uniqueness model,
281–286
LISREL-SIMPLIS program,
283–286
suggested reading, 291
Correlation coefficients, 4–5, 33–35,
42–46, See also Pearson
correlation coefficient
cause-effect relationships, 48–49
correction for attenuation, 39

Y102005.indb 496

Subject Index

curvilinear data and, 27
initial OLS estimates, 41
intraclass correlation, 319–320
level of measurement and, 34–36
matrix approach to computing,
432–434
missing data and, 38
model estimation problems, 217
nonlinear data and, 36–37
nonpositive definite matrices,
40–41
outliers and, 39
partial and part, See Partial and
part correlations
regression coefficient and, 126
suppressor variables and, 44
troubleshooting tips, 50
types, 34–35
Correlation computation using
variance and covariance, 47
Correlation matrix, 46–47, 425
confirmatory factor models and
decomposition, 169–170
model estimation and, 60, 202
path models and decomposition of,
151–152
Correlation versus covariance, 46–47
Covariance, path models, 144
Covariance matrix population data
LISREL matrix syntax, 355–359
PRELIS simulation, 346–352
SAS approach, 354–355
from specified model, 359–364
SPSS approach, 352–354
Covariance structure analysis, 189
Covariance structure modeling, 189
Covariance terms, 189–191
matrix notation, 378, 379–380
CP statistic, 135
Critical chi-square, 99
Critical N statistic, 41, 99
Critical value, 63, 64
Crossed research design, 307
Cross-validation, 42, 209, 219, 229–234
LISREL-SIMPLIS output, 231,
232–234
Cross-validation index (CVI), 230,
231–234

3/22/10 3:27:57 PM

497

Subject Index

Curvilinear data, 27, 36–37, 327, See
also Nonlinear models
troubleshooting tips, 30
D
Data imputation, 38
Data preparation, 212–214
checklist, 214
Data screening, 29
Data transformation, 28, 36, 340
Definite covariance matrices, 40–41, 50
Degrees of freedom
chi-square significance and, 85–86
expected cross-validation index,
230
MIMIC model, 294
model fit and, 75, 87, 89, 172, 184
model identification and, 58,
210, 294
noncentrality parameter and,
100, 102
parsimony NFI and, 90
partial and part correlation
significance, 34
power and, 93, 108, 110, 452
Dependent variables, 3, 180–181
multiple regression and prediction
and explanation, 127
Determinant, 427, 428, 430–431
Diagonal matrix, 431
Direct effects, path models, 144
Discrete variables, 19
Discriminant validity, 114, 182,
191, 277
Divergent validity, 182
Division of matrices, 427
Dow-Jones Index, 3
Dynamic factor model, 274–277
exercises, 288
LISREL-SIMPLIS program, 275–276
suggested reading, 290
E
Effect size, 108, 128
Endogenous latent variables, 181
EQS, 8

Y102005.indb 497

Equivalent models, 75, 211
Error terms, path models, 145
Error term zeta (ζ), 375, 376, 378
Error variance, 182
correlated, 303–304
observed variable measurement
error, 185
reliability coefficient, 183–184
Eta coefficient, 36
EXCEL, 258, 462
Exogenous latent variable, 181
Expectation maximization (EM),
20, 21
Expected cross-validation index
(ECVI), 230–231
Expected parameter change (EPC),
65–66, 155, 173, 205
Explanation and multiple regression,
127
Exploratory factor analysis (EFA),
116, 164
F
Factor analysis, 5, 33–34, 163–164,
See also Confirmatory factor
models
Factorial validity, 182
Factor loadings, 165, 183, See also
Confirmatory factor
analysis
bootstrap estimates, 238–240
matrix notation, 375, 377–378, 379
population model specification,
360
Factor model, 114, See also
Confirmatory factor models;
Measurement models
dynamic factor model, 274–277
second-order factor models,
271–274
Factor pattern matrix approach, 356,
358–359
F-distribution tables, 445–448
Fitted residuals, 64
Fitting functions, 60
Fixed parameters, 57, 150, 168, 201,
380

3/22/10 3:27:57 PM

498

Four-step approach, 115–116,
191–192
Free parameters, 57, 150, 168,
201, 380
F test, 34, 63, 128–129, 133
Full information estimation, 152
G
G*Power 3 software, 95–96, 102
Gamma (Γ) matrix, 375, 376, 379
Generalized least squares (GLS)
estimation, 60, 63, 86, 152
Goodness-of-fit index (GFI), 68, 76,
86–87, 154, 205, See also
Model-fit statistics
adjusted GFI (AGFI), 87, 89, 205
confirmatory factor models, 172
MIMIC model, 295, 297
GROUP command, 251, 255
Group mean differences between
observed variables, 259–262
H
Heywood case, 40, 92
Heywood variables, 217
Hierarchical multilevel models, See
Multilevel models
Hypothesis testing, 2, 93, 128
I
Identification problem, 56–57, 150,
200–201, 215
troubleshooting tips, 68
Identity matrix, 427, 432
Implied model, 41
Import data option, 15, 77
Imputation of Missing Values, 21–27
Incremental fit index (IFI), 76
Independence model, 41, 75
Independent variables, 3, 180–181
Indeterminacy, 57, 59
Indicator variables, See Observed
variables
Initial estimates, 40, 61, 92, See also
Start values

Y102005.indb 498

Subject Index

Intelligence assessment, 3, 180–181
Interaction effects, 327–328
testing in two-stage least-squares
estimate, 339
Interaction models, 7, 327–328
categorical variable approach,
328–331, 341
continuous variable approach,
330, 339
exercises, 366, 479–481
intercept only models, 328
intercept-slope models, 328
latent variable, 331–337, 339–341
LISREL matrix notation, 410–416
LISREL output, 416–421
LISREL program, 413–414
LISREL-SIMPLIS programs,
328–331, 335, 479–481
main effects model holding slopes
constant, 328–329
matrix specification, 414–416
multicollinearity, 340
path analysis and, 341
PRELIS output, 412–413
suggested reading, 368
two-stage least squares approach,
337–341
types of interaction effects, 327–328
Intercept model, 317
Intercept only models, 318–319, 323,
328, 477–478
Intercept-slope models, 328–329, 341–345
Intercept terms, 132, 138–139, 311, 335,
405, 437
Interval variables, 19, 35
Intraclass correlation coefficient,
319–320
Inverse of matrix, 427, 432
J
JKW model, 6
Just-identified model, 57, 131, 134–135
K
Kappa (κ), 405, 409
Kurtosis, 28, 36, 61

3/22/10 3:27:57 PM

Subject Index

L
Lambda X (Λ x) matrix, 358, 360, 361,
375, 379
Lambda Y (Λy) matrix, 356–357, 375,
379–381
Latent dependent variables, 180–181
matrix notation, 380–381
Latent growth curve models, 341–345
exercises, 366–367
LISREL-SIMPLIS program, 343
model modification, 344–345
suggested reading, 368
Latent Growth Curve Structured
Means Model, 345
Latent independent variables,
180, 181
matrix notation, 381–382
Latent variable interaction model,
331–337, See also Interaction
models
bootstrap estimates, 340
latent interaction variable
computation, 333–335
latent variable scores, 331–333, 341
LISREL matrix notation, 410–416
LISREL output, 416–421
LISREL program, 413–414
LISREL-SIMPLIS output, 335
matrix specification, 414–416
model modification, 336
PRELIS output, 412–413
problematic issues, 339–341
structural equation, 336
Latent variables, 2–3, 163, 180–181
factor analysis, 163–164, See also
Confirmatory factor
analysis
matrix notation, 374
observed variable selection, 183
origin and unit of measurement,
199
predicted by observed variables
(MIMIC models), 293–298
slope and intercept in latent
growth curve models,
342–345
standardization, 60

Y102005.indb 499

499

testing group mean differences,
259–262
variance-covariance terms, 189–191
Leptokurtic data, 28
Likelihood ratio (LR) test, 108
Limited information estimation, 152
Linearity assumptions, 27, 327
Linear regression models, See
Multiple regression models;
Regression models
LISREL, 6, 8, 13–14, 327, See
also LISREL-PRELIS;
LISREL-SIMPLIS
bootstrapping, 237–240
chi-square computation, 118–119
data entry, 14–18
expected parameter change
statistic, 65–66
historical development of, 6, 7
latent variable interaction model,
413–414, 416–421
multilevel modeling, 308–313
multiple samples program,
248–250
population covariance matrix,
355–359
using LISREL 8.8 student version,
8–10, 14
LISREL matrix notation, 373–379,
382–384
matrix program output example,
384–400
multiple-sample model, 404
path model, 400–403
structured means model,
405–410
see also Matrix approach to SEM
LISREL-PRELIS, 13, See also PRELIS
data entry, 14–18
missing data example, 21–27
mixture models, 301–302
non-normal data handling, 28–29
ordinal variables in, 36
outlier detection, 27
two-stage least squares analysis,
337
variables in, 298–299
LISREL-SIMPLIS, 6

3/22/10 3:27:57 PM

500

categorical variable interaction
model, 328–331
confirmatory factor model
program, 174–176
correlated traits-correlated
uniqueness model, 283–286
cross-validation index output,
232–234
data entry, 14–18
dynamic factor model, 275–276
expected cross-validation index
output, 231
interaction models, 328–331, 335,
479–481
latent growth curve program, 343
measurement model specification,
186, 199
MIMIC model, 294–298, 475–477
mixture models, 302–306
mixture model using polyserial
correlation matrix, 304–305
model-fit criteria program and
output example, 77–85
multiple group path model
analyses, 251–258
multiple regression analysis,
130, 132
multiple sample analysis, 224–229,
462–464
multitrait multimethod model,
280–281, 470–475
path model program, 156–158
population confirmatory factor
model, 363
population data simulation, 352
population model specification,
359
robust Satorra-Bentler scaled
chi-square, 306
second-order factor analysis,
272–274, 468–470
SEM program, 207–208
standardized solutions, 48
structural model specification,
187–188
structured means model, 260–262,
465–468
variance-covariance terms, 189–191

Y102005.indb 500

Subject Index

Listwise deletion, 20, 38
Logarithmic transformation, 36
Longitudinal data analysis, 341, 366
M
Main effects model for group
differences, 328–329
Mallow’s CP statistic, 135
Matching response pattern, 20, 21
Matrix approach to SEM
exercises, 422–423, 484–487
free, fixed, and constrained
parameters, 380
latent variable interaction models,
410–416
LISREL matrix command
language, 382–384
LISREL program output example,
384–400
matrix notation overview,
373–379
multiple-sample model, 404
path model, 400–403
population model specification,
361
SPSS program, 361
structural model equation,
373–374
Matrix beta (β), 374–375, 376, 379,
380
Matrix equation for structural model,
373–374, 376
Matrix gamma (Γ), 375, 376, 379
Matrix of cofactors, 427, 429–430
Matrix of minors, 427, 428–429
Matrix operations, 425, 432
addition and subtraction, 426
determinant, 427, 428, 430–431
division, 427–432
identity matrix, 432
inverse, 432
matrix definition, 425–426
multiple regression, 434–437
multiplication, 426–427
order of matrix, 426
Pearson correlation coefficient
computation, 432–434

3/22/10 3:27:57 PM

Subject Index

sums of squares and crossproducts, 433–434
transpose, 431
Matrix phi (Φ), 360, 361, 374, 378, 379
Matrix psi (Ψ), 375, 378, 379
Matrix sigma (Σ), See Sigma matrix
Maximum likelihood (ML)
estimation, 60, 61, 86, 152,
153, 204, 217
non-normal data handling, 62–63
Mean differences between observed
variables, 259–262
Mean substitution, 20, 38
Measurement error, 166, 182–184, 196
confirmatory factor model, 165–166
correction for attenuation, 39
correlated, 190, 197, 274, 345
matrix notation, 375, 379, 383
measurement models and, 185
parameter estimates and, 163
population model specification, 361
regression models and, 136–137
SEM and, 7
variance-covariance terms, 190
Measurement invariance, 116
Measurement models, 184–186,
196, 217, 271, See also
Confirmatory factor models
avoiding identification problems,
58–59
dynamic factor model, 274–277
four-step approach, 191–192
LISREL matrix notation, 405
LISREL-SIMPLIS program, 186, 199
matrix notation, 375, 376–377,
379–381
model fit, See Model fit
model modification and, 218, See
also Model modification
multicollinearity, 340
multiple samples, 224
multitrait multimethod model,
277–290
recommendations, 116
second-order factor models,
271–274
theoretical foundation, 212
two-step approach, 114–115, 191

Y102005.indb 501

501

Measurement scale, 18–19
correlation coefficients, 34–36
troubleshooting tips, 30, 50
Measurement validity issues, 163,
182–183
Mediating latent variables, 181
Meta-analysis, 212
Metropolitan Achievement Test,
184–185
MIMIC model, See Multiple
indicator-multiple cause
(MIMIC) models
Missing data, 20–21
correlation coefficients and, 38
LISREL-PRELIS example, 21–27
missing at random (MAR), 38
missing completely at random
(MCAR), 38
power and, 114
troubleshooting tips, 30, 50
Misspecified models, 56, 64–65,
130, 213, See also Model
specification
biased parameter estimates and,
56, 130, 213
confirmatory factor models,
169–174
model modification, 64–67, See also
Model modification
Mixture models, 35, 49, 298–307
exercises, 321–322
LISREL-PRELIS program,
301–302
LISREL-SIMPLIS programs,
302–306
model estimation and testing,
301–302
model modification, 303–306
model specification and
identification, 299–301
robust statistics, 305–306
suggested reading, 325
Model comparison, 88–89, 108–110
Model estimation, 59–63, 202–203,
210–211, 216–217
checklist, 217
confirmatory factor models,
169–170

3/22/10 3:27:57 PM

502

messy data problems, 217–218
MIMIC model, 294–298
mixture models, 301–302
multitrait multimethod model,
280–281
path models and, 151–154
regression model, 131–133
second-order factor models,
272–274
structured means model,
261–262
troubleshooting tips, 68
Model fit, 73, 85, 94, 217–218, 219
four-step approach, 115–116
ideal fit index, 117
model comparison, 88–89,
108–110
Monte Carlo methods, 364
parameter significance, 111–113
specification search, See
Specification search
structured means model, 261
two-step approach, 114–115
Model-fit criteria, 63, 74–77, 85–86,
See also Model-fit statistics;
specific indices
global fit measures, 75
LISREL-SIMPLIS example, 77–85
MIMIC model, 295, 297
R 2, 127–129
Model-fit statistics, 63, 68, 76–77, 85,
203–205, 211, 217–218, See also
specific indices
confirmatory factor models, 172
fitting functions, 60
model comparison, 88–89
model-fit criteria, 74–77
parsimony, 89–91
path models and, 154, 158–161
problems of SEM, 209
reporting, 91
sample size and, 41–42, 113–114
Model generating approach, 73
Model identification, 56–59, 200–202,
215–216
checklist, 216
confirmatory factor models,
167–169

Y102005.indb 502

Subject Index

free, fixed, and constrained
parameters, 57
methods for avoiding
identification problems,
58–59
MIMIC model, 294
mixture models, 299–301
multitrait multimethod model,
279
order and rank conditions, 58
path models and, 150–151
recommendations, 210–211
regression model, 131
second-order factor models,
271–272
structured means model, 259–260
troubleshooting tips, 68
Model modification, 64–67, 205–207,
211, 218–219
checklist, 218–219
confirmatory factor models,
173–175
latent growth curve model,
344–345
latent variable interaction model,
336
MIMIC model, 297
mixture model example, 303–306
path models and, 155–156
properly specified model, 66
regression analysis, 134–135
Model parsimony, 89–91, 115
Model specification, 55–56, 197–200,
213, 215, See also Misspecified
models; Specification search
checklist, 215
confirmatory factor models,
166–167
MIMIC model, 294
mixture models, 299–301
multitrait multimethod model, 279
path models and, 147–150
population covariance matrix,
359–364
regression model, 130, 135
second-order factor models,
271–272
structured means model, 259–260

3/22/10 3:27:58 PM

Subject Index

Model testing, 63, 203–205, 217–218,
See also Model-fit statistics;
Parameter estimates
checklist, 218
confirmatory factor models,
170–173
MIMIC model, 294–298
mixture models, 301–302
multitrait multimethod model,
280–281
path models and, 154
regression model, 133–134
second-order factor models,
272–274
structured means model, 261–262
Model validation, 218, 223, 241, 345,
See also Cross-validation
bootstrapping, 234–240
checklist, 219
cross-validation, 42
multitrait multimethod model,
277
replication using multiple samples,
223–229, 245–250
sample size and, 42
Modification indices, 65, 94, 155, 205
confirmatory factor models, 173
mixture models, 303
model generation approach, 73
power values, 111–113
Monte Carlo Markov Chain (MCMC),
20, 21
Monte Carlo methods, 335, 345–365
basic steps, 365
exercises, 368, 482–483
population covariance matrix from
specified model, 359–364
population data from specified
covariance matrix, 352–359,
482–483
PRELIS population data
simulation, 346–352
pseudo-random number
generation, 346
resources, 365
suggested reading, 368–369
Mplus, 8, 113–114
Multicollinearity, 340

Y102005.indb 503

503

Multilevel models, 7, 307–320
deviance statistic, 320
exercises, 323, 477–478
interpretation, 318–319
intraclass correlation, 319–320
LISREL resources, 308–313
null model, 308
PRELIS program, 313–318, 477–478
suggested reading, 325
variance decomposition, 308
Multiple correlation analysis (MCA),
127
Multiple correlation coefficient,
126–127
Multiple-group models, 7, 250–258
chi-square difference test, 258
exercises, 264–265
interaction effects, 328
LISREL-SIMPLIS path model
analyses, 251–258
separate group models, 251–254
similar group model, 255–258
suggested reading, 267
Multiple indicator-multiple cause
(MIMIC) models, 183,
293–298
exercises, 321, 475–477
goodness-of-fit criteria, 295, 297
LISREL-SIMPLIS program,
294–298, 475–477
model estimation and testing,
294–298
model modification, 297
model specification and
identification, 294
structural equations, 298
suggested reading, 324
Multiple linear regression, See
Multiple regression models
Multiple regression models, 125–129,
137–138, See also Regression
models
additive equation, 137–138
all-possible subset approach, 135
exercises, 289
LISREL-SIMPLIS program, 130,
132, 138–139
Mallow’s CP statistic, 135

3/22/10 3:27:58 PM

504

matrix approach, 434–437
measurement error and, 136–137
model identification and, 131
model modification, 134–135
model specification and, 130, 135
model testing, 133–134
path analysis and, 147–149
path models and, 143
prediction and explanation
applications, 127
R 2 index of fit, 127–129
robustness, 136, 138
Multiple samples, 223–229, 245–250
exercises, 263–264, 462–465
LISREL matrix notation, 404
LISREL program, 248–250
LISREL-SIMPLIS program,
224–229, 462–464
suggested reading, 267
Multiplication of matrices, 426
Multitrait multimethod model
(MTMM), 277–290
correlated uniqueness model,
281–286
exercises, 289–290, 470–475
LISREL-SIMPLIS programs,
280–281, 470–475
model estimation and testing,
280–281
model specification and
identification, 279
suggested reading, 290
Mx, 8
N
Nested models, 73, 211, 307
comparative fit index, 89
model comparison, 108
multilevel models, 307–320, See also
Multilevel models
parameter significance, 111–113
New option, 14–15
Nominal variables, 19, 35
Nomological validity, 114
Noncentrality index (NCI), 116
Noncentrality parameter (NCP), 75,
77, 94, 100, 102, 172

Y102005.indb 504

Subject Index

Nonignorable data, 38
Non-linear data, 27, 30, 36–37
troubleshooting tips, 50
Nonlinear models, 327–328, 341, See
also Interaction models
bootstrap estimates, 337, See also
Bootstrap
continuous variable approach,
330, 339
two-stage least squares approach,
337–341
Non-normality, 28
interaction models and, 340
model estimation and, 62
pseudo-random number
generation, 346
standard error estimation, 118
transformations, 28, 36
troubleshooting tips, 30
Nonpositive definite covariance
matrices, 40–41, 50
Nonrecursive structural models, 59
Normal distribution assumptions, 28,
61, 209, 217, 340
Normed fit index (NFI), 42, 76,
88–89
parsimony NFI (PNFI), 89, 90
relative NFI (RNFI), 114–115
O
Observed variables, 3–4, 180
categorical and continuous,
mixture models, 298–307
defining latent variables, 183
latent variable prediction (MIMIC
models), 293–298
measurement error and, 163, 196,
See also Measurement error
reference variables, 199
testing group mean differences,
259
Open option, 15
Order condition, 58, 150, 168, 201
Order of a matrix, 426
Ordinal variables, 19, 29, 35, 61–62
in LISREL-PRELIS, 36, 298
mixture models, 35

3/22/10 3:27:58 PM

Subject Index

Ordinary least squares (OLS)
estimation, 40–41, 59, 60
Origin of latent variable, 199
Outliers, 27, 92
correlation coefficients and, 39
troubleshooting tips, 30, 50
Output Options dialog box, 18
Over-identified model, 58, 211
P
Pairwise deletion, 20, 38
Parameter estimates, 63, 152–153,
217–218, See also Model
estimation
confirmatory factor models,
168–170
detecting specification
errors, 64
estimation methods, 152,
210–211, See also specific
methods
full and limited information,
152–153
initial estimates, 40, 61, 92
measurement error and, 163
misspecified model and bias, 56,
130, 213
model-fit criteria, 74
model identification and, 57, 150,
201, 215
Monte Carlo approach, 346,
352, 364
multiple group models, 250–258
multiple samples, 224, 245–250
outliers and, 92
parameter fit, 92–93
power values, 112–113
standard error estimation, 118
two-stage least squares approach,
337
Parameter significance, 92, 94,
111–113
Parsimony, 89–91
relative parsimony fit index, 115
Parsimony normed fit index (PNFI),
89, 90
Part correlation, 43–44

Y102005.indb 505

505

Partial and part correlations, 34,
42–46
significance testing, 34
standardized regression weights,
137
Partial regression coefficients, 126
Path coefficients, 147–149
Path diagram, confirmatory factor
model, 165
Path models, 5, 143–144
chi-square difference test, 258
chi-square test, 158
correlation matrix decomposition,
151–152
drawing conventions, 144–146
example, 144–146
interaction hypotheses, 341
LISREL matrix notation, 400–403
LISREL-SIMPLIS multiple group
analyses, 251–258
LISREL-SIMPLIS program, 156–158
model estimation and, 151–154
model-fit indices, 158–161
model identification and, 150–151
model modification, 155–156
model specification and, 147–150
model testing, 154
variables in, 3–4
Pattern matrix approach, 356, 358–359
Pearson correlation coefficient,
4–5, 29, 33–34, 35, See also
Correlation coefficients
cause-effect relationships, 48–49
correction for attenuation, 39
matrix approach to computing,
432–434
missing data and, 38
nonlinear data and, 37
outliers and, 39
Pearson product-moment correlation
coefficient, 35, See also
Pearson correlation
coefficient
path coefficients, 147–149
regression coefficient and, 126
Phi matrix (Φ), 360, 361, 374, 378, 379
Platykurtic data, 28
Polychoric correlations, 35, 62

3/22/10 3:27:58 PM

506

Polyserial correlations, 35, 62, 304–305
Population covariance matrix, 57,
150, 168, 201, 215, 346, 352,
361, See also Sigma matrix;
Variance–covariance matrix
confirmatory factor models, 360,
363–364
LISREL matrix syntax, 355–359
matrix operations, 361
PRELIS simulation, 346–352
SAS approach, 354–355
from specified model, 359–364
SPSS approach, 352–354, 482–483
Power, 93–99
G*Power 3 software, 95–96, 102
missing data and, 114
model comparison, 108–110
parameter estimates, 112–113
RMSEA and, 106–107
RMSEA and effect size, 108–110
SAS syntax, 95, 106, 111–113
SPSS syntax, 95
Prediction and multiple regression,
127
Predictive validity assessment, 191
PRELIS, 49, 346, See also
LISREL-PRELIS
bootstrapping, 234–240
latent variable interaction model,
331–335, 412–413
model estimation and, 62
multilevel modeling, 313–318,
477–478
population data simulation,
346–352
system file, 17
two-stage least squares analysis,
337–338
variables in, 19–20, 29, 62, 298–299
Probit data transformation, 28, 36
Properly specified model, 66
Pseudo-random number generation,
346
Psi matrix (Ψ), 375, 378, 379
Q
Q, 158, 160

Y102005.indb 506

Subject Index

R
R 2, 127–129, 132–137
correction for measurement error,
136–137
path model-fit index, 159
Ramona, 8
Random sample, SPSS, 268
Rank comparison, 19
Rank condition, 58, 202
Ratio variables, 19, 35
Reciprocal transformation, 36
Recursive structural models, 59
Reference variables, 199, 296
Regression, 33–34
Regression imputation, 20
Regression models, 4, 125–129, See also
Multiple regression models
LISREL-SIMPLIS program, 130,
132, 138–139
model estimation and, 131–133
model identification and, 131
model modification, 134–135
model specification and, 130, 135
model testing, 133–134
saturated just-identified models,
131, 134
theoretical framework, 134
variable measurement error,
136–137
variables in, 3
Regression weight correction for
attenuation, 137
Relative fit index (RFI), 76
Relative noncentrality index (RNI), 89
Relative normed fit index (RNFI),
114–115
Relative parsimony fit index, 115
Relative parsimony ratio, 115
Reliability, 163, 182
correction for attenuation, 39, 50
Reliability coefficient, 183–184, 277
multitrait multimethod model, 277
Repeated measures ANOVA, 341–342
latent growth curve model,
341–345
Replication, 219
multiple samples, 223–229, 245–250

3/22/10 3:27:58 PM

Subject Index

Reporting SEM research, 209
checklist, 211–212
data preparation, 212–214
model estimation checklist, 217
model-fit indices, 91
model identification checklist,
216
model modification checklist,
218–219
model specification checklist, 215
model testing checklist, 218
model validation checklist, 219
recommendations, 209–210
Residual matrix, 64, 87–88, 155,
173, 205
chi-square value and, 75
Restriction of range, 19–20
troubleshooting tips, 30, 50
Root-mean-square error of
approximation (RMSEA),
76–77, 154, 203
confirmatory factor models, 172
effect size and power, 108–110
model-fit criteria, 74
reporting, 91
sample size and, 42
SAS syntax, 106
Root-mean-square residual index
(RMR), 87
S
Sample matrix description, 212
Sample size, 41–42, 93, 114
chi-square and, 41, 86, 99–100,
211
critical N statistic, 41, 99
minimum satisfactory, 42
model-fit indices and, 113–114
model validation and, 42
parameter fit and, 92
problems of SEM, 209
rule of thumb, 211
SAS syntax, 101–104
troubleshooting tips, 40–41
Sample variance-covariance matrix,
See Variance-covariance
matrix

Y102005.indb 507

507

SAS syntax
effect size, RMSEA, and power,
110
population covariance matrix,
354–355
power, 95
power for parameter modification
indices, 111–113
RMSEA and power, 106
sample size, 101–104
Satorra-Bentler scaled chi-square, 62,
119, 305–306
Saturated model, 41, 75, 131, 134
Scalar matrix multiplication, 427
Scatterplot, 27, 36
Second-order factor models, 271–274
exercises, 287, 468–470
LISREL-SIMPLIS program,
272–274, 468–470
model estimation and testing,
272–274
model specification and
identification, 271–272
suggested reading, 290
Sepath, 8
SEPATH, 60
Sigma matrix (Σ), 57, 59–60, 75,
150, 168, 191, 201, 215,
361, 378–379, See also
Population covariance
matrix; Variance–
covariance matrix
Significance, See Statistical
significance
SIMPLIS, 6, See also LISREL-SIMPLIS
Skewness, 28, 36
Slope and intercept parameters, latent
growth model, 341–345
Specification error, 56, 113, 130
detection methods, 64
path models and, 155
recommended procedure, 67
Specification search, 64–67, 73–74,
205–207, 211
confirmatory factor models, 173
troubleshooting tips, 68
Split-sample cross-validation,
229–232

3/22/10 3:27:58 PM

508

SPSS syntax
MATRIX procedure, 362
population covariance matrix,
352–354, 482–483
power, 95
RMSEA and power, 107
Select Cases: Random Sample,
268
special data file types, 212
Squared multiple correlation
coefficient (R 2), 127–129,
132–137, 159
Square root transformation, 36
Standard error
alternative model identification
and, 94
bootstrap estimates, 234
computation, 60, 63, 118
computation, Monte Carlo
methods, 364
Standardization of latent
variables, 60
Standardized partial regression
coefficients, 147
Standardized partial regression
weights, 137
Standardized regression coefficient
(β), 126, 133–134
Standardized residual matrix,
64–65, 155, 173, 202–203,
205
Standardized root-mean-square
residual index, 87, 91, 154
Standardized variables, 47–48
Standardized z scores, 64–65, 126
Stanford-Binet Intelligence Score, 180,
181, 184
Start values, 61, See also Initial
estimates
nonpositive definite matrices, 40
two-stage least squares approach,
337
Statistical significance, 85, 94
chi-square test, 75, See also
Chi-square
confirmatory factor model
approach, 164

Y102005.indb 508

Subject Index

F test, 34, 63, 128–129, 133
hypothesis testing, 93
missing data and, 38
model-fit criteria, 74–75, See also
Chi-square; Model fit;
Model-fit criteria
model modification and, 155,
173, 205
model testing, 48
model testing and, 217
parameter, 92, 94, 111–113
partial and part correlation, 34
path model fit, 158
R 2, 128
r distribution table, 442
specification search, 64
t distribution table, 441
Structural equation matrices, 373–379,
See also Matrix approach to
SEM
Structural equation model
development, 179, 195,
See also Measurement
models; Structural model
example, 195–197
four-step approach, 191–192
LISREL-SIMPLIS program,
207–208
model estimation, 202–203
model identification, 200–202
model modification, 205–207
model specification, 197–200
model testing, 203–205
two-step approach, 191
variance and covariance terms,
189–191, 200
Structural Equation Modeling
(journal), 6
Structural equation modeling (SEM),
2–4, 179
history of, 4–6
reasons for using, 6–7
reporting, See Reporting SEM
research
sample size requirements, 211,
See also Sample size
shortcomings, 209

3/22/10 3:27:58 PM

509

Subject Index

“10 commandments”, 210
Structural equation modeling (SEM),
basic components, 55, 73,
See also Model estimation;
Model identification;
Model modification; Model
specification; Model testing
Structural equation modeling (SEM)
software programs, 7–8,
See also LISREL; LISRELSIMPLIS; PRELIS
LISREL 8.8 student version, 8
Structural model, 114, 186–188, 200,
212, 217
LISREL-SIMPLIS program, 187–188
matrix equation, 373–374, 376
MIMIC, 298
model modification and, 218
multiple samples, 224
recommendations, 116
Structure coefficients, 187, 197
Structured means models, 259–262
exercises, 265–266, 465–468
LISREL matrix notation, 405–410
LISREL-SIMPLIS programs,
260–262, 465–468
model estimation and testing,
261–262
model fit, 261
model specification and
identification, 259–260
suggested reading, 267
Subtraction of matrices, 426
Sums of squares and cross-products
(SSCP) matrix, 433–434
Suppressor variables, 44

model fit, See Model-fit criteria
model specification and, 56, 213,
See also Model specification
regression models and, 134
replication, 219, 223–229
validation, See Cross-validation;
Model validation
Theta-delta (Θδ) matrix, 360–361, 375,
378, 379
Theta-epsilon (Θε) matrix, 375, 378,
379
Transposed matrix, 427, 431
Triangular matrix, 431
Troubleshooting tips, 30, 50, 68
True score correlation, 39
t rule, 58
t statistic, 74, 205
distribution for given probability
(table), 441
Tucker-Lewis index (TLI), 76, 88,
113
Two-stage least-squares (TSLS)
estimates, 92, 337–341
LISREL-PRELIS program, 337
testing interaction effects, 339
Two-step model-building approach,
114–115, 191

Tabu, 66, 73–74
Tau (τ), 361, 405
Tetrachoric correlations, 29, 35
TETRAD, 66
Theoretical models, 2, 143, 210, 213
identification problem, 57, 150,
201, 215, See also Model
identification

Validation of model, See Model
validation
Validity issues in measurement,
163, 182–183, 191, See also
Measurement models
Variables, 2–3, See also specific types
standardized and unstandardized,
47–48

Y102005.indb 509

U
Under-identified model, 57
Unit of measurement, 199
Unstandardized coefficients,
47–48
Unweighted least squares (ULS)
estimation, 60, 86, 152

3/22/10 3:27:59 PM

510

Variance-covariance matrix, 46–47,
189, 191, 202, See also
Population covariance
matrix; Sigma matrix
in LISREL-PRELIS, 298–299
matrix approach to correlation
coefficient computation,
434
matrix notation, 374–375
model estimation and, 61, 202,
210–211
nonpositive definite, 40–41
Variance-covariance terms, 189–191,
200
Variance decomposition, 308

Subject Index

Weighted-least squares (WLS)
estimation, 60, 61, 63, 100
W path model-fit index, 160
X
X (independent variable), 180, 181
X and Y cause-effect relationships,
48–49
X scores, 4
Y
Y (dependent variable), 180, 181
Y scores, 4
Z

W
Wechsler Intelligence Scale for
Children — Revised
(WISC-R), 3, 180, 181, 184

Y102005.indb 510

Zeta error term (ζ), 375, 376, 378
z scores, 64–65, 126
z scores, 155
table, 440

3/22/10 3:27:59 PM

Source Exif Data:

File Type                       : PDF
File Type Extension             : pdf
MIME Type                       : application/pdf
PDF Version                     : 1.7
Linearized                      : Yes
Author                          : 
Create Date                     : 2010:03:22 15:24:34+05:30
EBX PUBLISHER                   : Taylor & Francis Ltd
Modify Date                     : 2017:11:02 14:59:13-04:00
Title                           : A Beginner's Guide to Structural Equation Modeling
Has XFA                         : No
XMP Toolkit                     : Adobe XMP Core 5.6-c015 81.157285, 2014/12/12-00:43:15
Producer                        : Adobe PDF Library 8.0
Trapped                         : False
Metadata Date                   : 2017:11:02 14:59:13-04:00
Creator Tool                    : Adobe InDesign CS3 (5.0)
Thumbnail Format                : JPEG
Thumbnail Width                 : 256
Thumbnail Height                : 256
Thumbnail Image                 : (Binary data 4432 bytes, use -b option to extract)
Instance ID                     : uuid:c92986ce-0e29-42a7-a41e-2eb8c2ee24cb
Rendition Class                 : proof:pdf
Document ID                     : adobe:docid:indd:9eba88ef-119c-11df-bd01-c880a0b6aaf4
Derived From Instance ID        : 83224af3-119b-11df-9134-d8ea2295151e
Derived From Document ID        : adobe:docid:indd:83224af2-119b-11df-9134-d8ea2295151e
Manifest Link Form              : ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream, ReferenceStream
Manifest Placed X Resolution    : 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00
Manifest Placed Y Resolution    : 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00, 72.00
Manifest Placed Resolution Unit : Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches, Inches
Manifest Reference Instance ID  : uuid:1bdc24e0-f480-4557-9850-388dda556350, uuid:198399ce-8117-42c8-9c55-726b1692aab3, uuid:05e29f53-d7de-4f3d-968a-7425054ec2fa, uuid:A8A3C8C2CBE7DE11A31D91CFBD4AF261, uuid:e6500bf2-e7cb-11de-8b71-e66145b7d767, uuid:32b5bf9f-e7cc-11de-8b71-e66145b7d767, uuid:8dda6c12-e7d0-11de-8b71-e66145b7d767, uuid:D7774B4B38EADE1190719E56733AE11F, uuid:045bf3fd-e7cd-11de-8b71-e66145b7d767, uuid:bcccd73c-e7d0-11de-8b71-e66145b7d767, uuid:4D4E356738EADE1190719E56733AE11F, uuid:ba355d16-e7d9-11de-8b71-e66145b7d767, uuid:03124193-e7de-11de-8b71-e66145b7d767, uuid:1c8a61f9-e7de-11de-8b71-e66145b7d767, uuid:0312418b-e7de-11de-8b71-e66145b7d767, uuid:0E2C198138EADE1190719E56733AE11F, uuid:cb462e4a-e7dd-11de-8b71-e66145b7d767, uuid:B0838A8B38EADE1190719E56733AE11F, uuid:B657868F38EADE1190719E56733AE11F, uuid:312fef40-e7de-11de-8b71-e66145b7d767, uuid:e70b2d16-e7dd-11de-8b71-e66145b7d767, uuid:0f07d05a-e7de-11de-8b71-e66145b7d767, uuid:F7FAA78260EADE11A8DAF3FDEF3F6791, uuid:A791A4F83AEADE11A8DAF3FDEF3F6791, uuid:646492ED3CEADE11A8DAF3FDEF3F6791, uuid:656492ED3CEADE11A8DAF3FDEF3F6791, uuid:A891A4F83AEADE11A8DAF3FDEF3F6791, uuid:77ad1e11-e7e1-11de-8b71-e66145b7d767, uuid:666492ED3CEADE11A8DAF3FDEF3F6791, uuid:8e12db82-e7e1-11de-8b71-e66145b7d767, uuid:232f258c-e7e1-11de-8b71-e66145b7d767, uuid:6a23f880-e7e1-11de-8b71-e66145b7d767, uuid:F0FAA78260EADE11A8DAF3FDEF3F6791, uuid:96cd0321-e7e1-11de-8b71-e66145b7d767, uuid:D2F85F0E39EADE1190719E56733AE11F, uuid:502dd325-e7e1-11de-8b71-e66145b7d767, uuid:e8af4c72-e7e1-11de-8b71-e66145b7d767, uuid:114a381c-e7e2-11de-8b71-e66145b7d767, uuid:db52caf9-e7e1-11de-8b71-e66145b7d767, uuid:db52cb01-e7e1-11de-8b71-e66145b7d767, uuid:f940fa5b-e7e1-11de-8b71-e66145b7d767, uuid:114a3826-e7e2-11de-8b71-e66145b7d767, uuid:696492ED3CEADE11A8DAF3FDEF3F6791, uuid:686492ED3CEADE11A8DAF3FDEF3F6791, uuid:AE55C63CE5E7DE11A31D91CFBD4AF261, uuid:BABA77EAE0E7DE11A31D91CFBD4AF261, uuid:6B6492ED3CEADE11A8DAF3FDEF3F6791, uuid:EC8854A039EADE11A8DAF3FDEF3F6791, uuid:B455C63CE5E7DE11A31D91CFBD4AF261, uuid:B655C63CE5E7DE11A31D91CFBD4AF261, uuid:3EB35EF6E6E7DE11A31D91CFBD4AF261, uuid:ED8854A039EADE11A8DAF3FDEF3F6791, uuid:B198BE9D42EADE11A8DAF3FDEF3F6791, uuid:6A6492ED3CEADE11A8DAF3FDEF3F6791, uuid:EF8854A039EADE11A8DAF3FDEF3F6791, uuid:1E50EF8A0D8011DF8F2AC471904A1ED9, uuid:B298BE9D42EADE11A8DAF3FDEF3F6791, uuid:F18854A039EADE11A8DAF3FDEF3F6791, uuid:1ed01708-e7f4-11de-8b71-e66145b7d767, uuid:1ed01710-e7f4-11de-8b71-e66145b7d767, uuid:57A59975F3E7DE11A31D91CFBD4AF261, uuid:F1FAA78260EADE11A8DAF3FDEF3F6791, uuid:c531f6be-e7f4-11de-8b71-e66145b7d767, uuid:b57e71cd-e7f4-11de-8b71-e66145b7d767, uuid:d06e577f-e7f4-11de-8b71-e66145b7d767, uuid:c531f6c6-e7f4-11de-8b71-e66145b7d767, uuid:9f99d378-e7f4-11de-8b71-e66145b7d767, uuid:6D6492ED3CEADE11A8DAF3FDEF3F6791, uuid:F2FAA78260EADE11A8DAF3FDEF3F6791, uuid:1E50EF8C0D8011DF8F2AC471904A1ED9, uuid:3E0633563FEADE11A8DAF3FDEF3F6791, uuid:A8A159E33DEADE11A8DAF3FDEF3F6791, uuid:d7ed0eec-e7f4-11de-8b71-e66145b7d767, uuid:8b0a66f2-e7f7-11de-8b71-e66145b7d767, uuid:b57fe6cf-e7f7-11de-8b71-e66145b7d767, uuid:6E6492ED3CEADE11A8DAF3FDEF3F6791, uuid:80E0F4EAF8E7DE11A31D91CFBD4AF261, uuid:BEEB398BFCE7DE11A31D91CFBD4AF261, uuid:7af5a22f-e7f7-11de-8b71-e66145b7d767, uuid:9F6833740FE8DE11A31D91CFBD4AF261, uuid:420633563FEADE11A8DAF3FDEF3F6791, uuid:430633563FEADE11A8DAF3FDEF3F6791, uuid:5EBDE64214E8DE11A31D91CFBD4AF261, uuid:F3FAA78260EADE11A8DAF3FDEF3F6791, uuid:62BDE64214E8DE11A31D91CFBD4AF261, uuid:202C4DE715E8DE11A31D91CFBD4AF261, uuid:222C4DE715E8DE11A31D91CFBD4AF261, uuid:F4FAA78260EADE11A8DAF3FDEF3F6791, uuid:E435B91427EADE11A8DAF3FDEF3F6791, uuid:E635B91427EADE11A8DAF3FDEF3F6791, uuid:f0c8223c-e7fd-11de-8b71-e66145b7d767, uuid:063CACA042EADE1190719E56733AE11F, uuid:b45578ee-e7fd-11de-8b71-e66145b7d767, uuid:056197d6-e7fe-11de-8b71-e66145b7d767, uuid:450633563FEADE11A8DAF3FDEF3F6791, uuid:460633563FEADE11A8DAF3FDEF3F6791, uuid:fb6e4233-e7fd-11de-8b71-e66145b7d767, uuid:470633563FEADE11A8DAF3FDEF3F6791, uuid:4ced9532-e7fe-11de-8b71-e66145b7d767, uuid:1d8773e8-e7fe-11de-8b71-e66145b7d767, uuid:e97417e9-e7fd-11de-8b71-e66145b7d767, uuid:d3394cce-e7fd-11de-8b71-e66145b7d767, uuid:a3c68665-e7fd-11de-8b71-e66145b7d767, uuid:d3394cd6-e7fd-11de-8b71-e66145b7d767, uuid:2c5c7099-e7fe-11de-8b71-e66145b7d767, uuid:3aadd71d-e7fe-11de-8b71-e66145b7d767, uuid:cb2ce79b-e7fd-11de-8b71-e66145b7d767, uuid:2c5c7091-e7fe-11de-8b71-e66145b7d767, uuid:34341e18-e7fe-11de-8b71-e66145b7d767, uuid:480633563FEADE11A8DAF3FDEF3F6791, uuid:A898BE9D42EADE11A8DAF3FDEF3F6791, uuid:15e12c77-e7fe-11de-8b71-e66145b7d767, uuid:b45578f6-e7fd-11de-8b71-e66145b7d767, uuid:C6DC71A52AEADE11A8DAF3FDEF3F6791, uuid:b30b0d58-e7fe-11de-8b71-e66145b7d767, uuid:d99f1038-e7fe-11de-8b71-e66145b7d767, uuid:08032454-e7ff-11de-8b71-e66145b7d767, uuid:EA09EEB842EADE1190719E56733AE11F, uuid:1dfb2f3a-e7ff-11de-8b71-e66145b7d767, uuid:97ED0FD72EEADE11A8DAF3FDEF3F6791, uuid:9AED0FD72EEADE11A8DAF3FDEF3F6791, uuid:A998BE9D42EADE11A8DAF3FDEF3F6791, uuid:AA98BE9D42EADE11A8DAF3FDEF3F6791, uuid:FE82577E4CEADE1190719E56733AE11F, uuid:2c304329-e7ff-11de-8b71-e66145b7d767, uuid:F5FAA78260EADE11A8DAF3FDEF3F6791, uuid:BD75719F4CEADE1190719E56733AE11F, uuid:5BDA46A64CEADE1190719E56733AE11F, uuid:36eb4aae-e7ff-11de-8b71-e66145b7d767, uuid:00d73ad5-e7ff-11de-8b71-e66145b7d767, uuid:e155aea1-e7fe-11de-8b71-e66145b7d767, uuid:f90b6790-e7fe-11de-8b71-e66145b7d767, uuid:0803244c-e7ff-11de-8b71-e66145b7d767, uuid:17bad6cd-e7ff-11de-8b71-e66145b7d767, uuid:d022d15f-e7fe-11de-8b71-e66145b7d767, uuid:2F356CBE33EADE11A8DAF3FDEF3F6791, uuid:31356CBE33EADE11A8DAF3FDEF3F6791, uuid:1E50EF8E0D8011DF8F2AC471904A1ED9, uuid:F6FAA78260EADE11A8DAF3FDEF3F6791, uuid:1E50EF900D8011DF8F2AC471904A1ED9, uuid:AAA159E33DEADE11A8DAF3FDEF3F6791, uuid:9F91A4F83AEADE11A8DAF3FDEF3F6791, uuid:A191A4F83AEADE11A8DAF3FDEF3F6791, uuid:A391A4F83AEADE11A8DAF3FDEF3F6791, uuid:A091A4F83AEADE11A8DAF3FDEF3F6791, uuid:A691A4F83AEADE11A8DAF3FDEF3F6791, uuid:136b5f5d-e7d2-11de-8b71-e66145b7d767, uuid:2f38a3c4-e7d2-11de-8b71-e66145b7d767, uuid:B0A3C8C2CBE7DE11A31D91CFBD4AF261, uuid:A591A4F83AEADE11A8DAF3FDEF3F6791, uuid:A491A4F83AEADE11A8DAF3FDEF3F6791, uuid:F68854A039EADE11A8DAF3FDEF3F6791, uuid:400633563FEADE11A8DAF3FDEF3F6791, uuid:66e84c65-e7d5-11de-8b71-e66145b7d767, uuid:8128cc14-e7d5-11de-8b71-e66145b7d767, uuid:E4736EB25BEADE1190719E56733AE11F, uuid:da98d55a-e7d4-11de-8b71-e66145b7d767, uuid:4550d86b-e7d5-11de-8b71-e66145b7d767, uuid:4550d863-e7d5-11de-8b71-e66145b7d767, uuid:C7B549D90D8211DF8F2AC471904A1ED9
Manifest Reference Document ID  : xmp.did:6FE6109127DF1168837DD81327823FB2, adobe:docid:indd:b930c155-2ffa-11de-8f60-c1211d7f0819, xmp.did:D123DEE7200911689F77C55174C0E1F8, uuid:A7A3C8C2CBE7DE11A31D91CFBD4AF261, adobe:docid:photoshop:e6500bf1-e7cb-11de-8b71-e66145b7d767, adobe:docid:photoshop:32b5bf9e-e7cc-11de-8b71-e66145b7d767, adobe:docid:photoshop:8dda6c11-e7d0-11de-8b71-e66145b7d767, adobe:docid:photoshop:8dda6c19-e7d0-11de-8b71-e66145b7d767, adobe:docid:photoshop:045bf3fc-e7cd-11de-8b71-e66145b7d767, adobe:docid:photoshop:bcccd73b-e7d0-11de-8b71-e66145b7d767, adobe:docid:photoshop:af78ed3e-e7d0-11de-8b71-e66145b7d767, adobe:docid:photoshop:ba355d15-e7d9-11de-8b71-e66145b7d767, adobe:docid:photoshop:03124192-e7de-11de-8b71-e66145b7d767, adobe:docid:photoshop:1c8a61f8-e7de-11de-8b71-e66145b7d767, adobe:docid:photoshop:0312418a-e7de-11de-8b71-e66145b7d767, adobe:docid:photoshop:d9478490-e7dd-11de-8b71-e66145b7d767, adobe:docid:photoshop:cb462e49-e7dd-11de-8b71-e66145b7d767, adobe:docid:photoshop:2873f24e-e7dd-11de-8b71-e66145b7d767, adobe:docid:photoshop:312fef47-e7de-11de-8b71-e66145b7d767, adobe:docid:photoshop:1c8a6200-e7de-11de-8b71-e66145b7d767, adobe:docid:photoshop:e70b2d15-e7dd-11de-8b71-e66145b7d767, adobe:docid:photoshop:0f07d059-e7de-11de-8b71-e66145b7d767, uuid:A9A3C8C2CBE7DE11A31D91CFBD4AF261, uuid:2A3D6D36D7E7DE11A31D91CFBD4AF261, uuid:516535CDDEE7DE11A31D91CFBD4AF261, uuid:536535CDDEE7DE11A31D91CFBD4AF261, uuid:556535CDDEE7DE11A31D91CFBD4AF261, adobe:docid:photoshop:77ad1e10-e7e1-11de-8b71-e66145b7d767, uuid:5A6535CDDEE7DE11A31D91CFBD4AF261, adobe:docid:photoshop:8e12db81-e7e1-11de-8b71-e66145b7d767, adobe:docid:photoshop:232f258b-e7e1-11de-8b71-e66145b7d767, adobe:docid:photoshop:6a23f87f-e7e1-11de-8b71-e66145b7d767, uuid:586535CDDEE7DE11A31D91CFBD4AF261, adobe:docid:photoshop:96cd0320-e7e1-11de-8b71-e66145b7d767, adobe:docid:photoshop:77ad1e18-e7e1-11de-8b71-e66145b7d767, adobe:docid:photoshop:502dd324-e7e1-11de-8b71-e66145b7d767, adobe:docid:photoshop:e8af4c71-e7e1-11de-8b71-e66145b7d767, adobe:docid:photoshop:f940fa62-e7e1-11de-8b71-e66145b7d767, adobe:docid:photoshop:db52caf8-e7e1-11de-8b71-e66145b7d767, adobe:docid:photoshop:db52cb00-e7e1-11de-8b71-e66145b7d767, adobe:docid:photoshop:f940fa5a-e7e1-11de-8b71-e66145b7d767, adobe:docid:photoshop:114a3825-e7e2-11de-8b71-e66145b7d767, uuid:B4BA77EAE0E7DE11A31D91CFBD4AF261, uuid:B6BA77EAE0E7DE11A31D91CFBD4AF261, uuid:BBBA77EAE0E7DE11A31D91CFBD4AF261, uuid:B9BA77EAE0E7DE11A31D91CFBD4AF261, uuid:AF55C63CE5E7DE11A31D91CFBD4AF261, uuid:B155C63CE5E7DE11A31D91CFBD4AF261, uuid:B355C63CE5E7DE11A31D91CFBD4AF261, uuid:B555C63CE5E7DE11A31D91CFBD4AF261, uuid:B755C63CE5E7DE11A31D91CFBD4AF261, uuid:3FB35EF6E6E7DE11A31D91CFBD4AF261, uuid:41B35EF6E6E7DE11A31D91CFBD4AF261, uuid:47B35EF6E6E7DE11A31D91CFBD4AF261, uuid:44B35EF6E6E7DE11A31D91CFBD4AF261, uuid:1E50EF890D8011DF8F2AC471904A1ED9, uuid:4FA59975F3E7DE11A31D91CFBD4AF261, uuid:301BC4BDE7E7DE11A31D91CFBD4AF261, adobe:docid:photoshop:2319de89-e7f2-11de-8b71-e66145b7d767, adobe:docid:photoshop:1ed0170f-e7f4-11de-8b71-e66145b7d767, uuid:55A59975F3E7DE11A31D91CFBD4AF261, uuid:58A59975F3E7DE11A31D91CFBD4AF261, adobe:docid:photoshop:b57e71d4-e7f4-11de-8b71-e66145b7d767, adobe:docid:photoshop:b57e71cc-e7f4-11de-8b71-e66145b7d767, adobe:docid:photoshop:d06e577e-e7f4-11de-8b71-e66145b7d767, adobe:docid:photoshop:c531f6c5-e7f4-11de-8b71-e66145b7d767, adobe:docid:photoshop:9f99d377-e7f4-11de-8b71-e66145b7d767, uuid:B10936EDF5E7DE11A31D91CFBD4AF261, uuid:B70936EDF5E7DE11A31D91CFBD4AF261, uuid:1E50EF8B0D8011DF8F2AC471904A1ED9, uuid:81E0F4EAF8E7DE11A31D91CFBD4AF261, uuid:84E0F4EAF8E7DE11A31D91CFBD4AF261, adobe:docid:photoshop:d7ed0eeb-e7f4-11de-8b71-e66145b7d767, adobe:docid:photoshop:8b0a66f1-e7f7-11de-8b71-e66145b7d767, adobe:docid:photoshop:b57fe6ce-e7f7-11de-8b71-e66145b7d767, uuid:BA0936EDF5E7DE11A31D91CFBD4AF261, uuid:7FE0F4EAF8E7DE11A31D91CFBD4AF261, uuid:88E0F4EAF8E7DE11A31D91CFBD4AF261, adobe:docid:photoshop:7af5a22e-e7f7-11de-8b71-e66145b7d767, uuid:9E6833740FE8DE11A31D91CFBD4AF261, uuid:A26833740FE8DE11A31D91CFBD4AF261, uuid:5ABDE64214E8DE11A31D91CFBD4AF261, uuid:5DBDE64214E8DE11A31D91CFBD4AF261, uuid:5FBDE64214E8DE11A31D91CFBD4AF261, uuid:61BDE64214E8DE11A31D91CFBD4AF261, uuid:63BDE64214E8DE11A31D91CFBD4AF261, uuid:212C4DE715E8DE11A31D91CFBD4AF261, uuid:232C4DE715E8DE11A31D91CFBD4AF261, uuid:E335B91427EADE11A8DAF3FDEF3F6791, uuid:E535B91427EADE11A8DAF3FDEF3F6791, adobe:docid:photoshop:f0c8223b-e7fd-11de-8b71-e66145b7d767, adobe:docid:photoshop:056197cd-e7fe-11de-8b71-e66145b7d767, adobe:docid:photoshop:a3c6866c-e7fd-11de-8b71-e66145b7d767, adobe:docid:photoshop:056197d5-e7fe-11de-8b71-e66145b7d767, uuid:E735B91427EADE11A8DAF3FDEF3F6791, uuid:E935B91427EADE11A8DAF3FDEF3F6791, adobe:docid:photoshop:fb6e4232-e7fd-11de-8b71-e66145b7d767, uuid:BCDC71A52AEADE11A8DAF3FDEF3F6791, adobe:docid:photoshop:3aadd724-e7fe-11de-8b71-e66145b7d767, adobe:docid:photoshop:1d8773e7-e7fe-11de-8b71-e66145b7d767, adobe:docid:photoshop:e97417e8-e7fd-11de-8b71-e66145b7d767, adobe:docid:photoshop:d3394ccd-e7fd-11de-8b71-e66145b7d767, adobe:docid:photoshop:a3c68664-e7fd-11de-8b71-e66145b7d767, adobe:docid:photoshop:d3394cd5-e7fd-11de-8b71-e66145b7d767, adobe:docid:photoshop:2c5c7098-e7fe-11de-8b71-e66145b7d767, adobe:docid:photoshop:3aadd71c-e7fe-11de-8b71-e66145b7d767, adobe:docid:photoshop:cb2ce79a-e7fd-11de-8b71-e66145b7d767, adobe:docid:photoshop:2c5c7090-e7fe-11de-8b71-e66145b7d767, adobe:docid:photoshop:34341e17-e7fe-11de-8b71-e66145b7d767, uuid:BFDC71A52AEADE11A8DAF3FDEF3F6791, uuid:C2DC71A52AEADE11A8DAF3FDEF3F6791, adobe:docid:photoshop:15e12c76-e7fe-11de-8b71-e66145b7d767, adobe:docid:photoshop:b45578f5-e7fd-11de-8b71-e66145b7d767, uuid:C5DC71A52AEADE11A8DAF3FDEF3F6791, adobe:docid:photoshop:b30b0d57-e7fe-11de-8b71-e66145b7d767, adobe:docid:photoshop:d99f1037-e7fe-11de-8b71-e66145b7d767, adobe:docid:photoshop:08032453-e7ff-11de-8b71-e66145b7d767, adobe:docid:photoshop:d022d156-e7fe-11de-8b71-e66145b7d767, adobe:docid:photoshop:1dfb2f39-e7ff-11de-8b71-e66145b7d767, uuid:96ED0FD72EEADE11A8DAF3FDEF3F6791, uuid:98ED0FD72EEADE11A8DAF3FDEF3F6791, uuid:9BED0FD72EEADE11A8DAF3FDEF3F6791, uuid:9DED0FD72EEADE11A8DAF3FDEF3F6791, adobe:docid:photoshop:9c345014-e7fe-11de-8b71-e66145b7d767, adobe:docid:photoshop:2c304328-e7ff-11de-8b71-e66145b7d767, uuid:9FED0FD72EEADE11A8DAF3FDEF3F6791, adobe:docid:photoshop:2c304320-e7ff-11de-8b71-e66145b7d767, adobe:docid:photoshop:f90b6797-e7fe-11de-8b71-e66145b7d767, adobe:docid:photoshop:36eb4aad-e7ff-11de-8b71-e66145b7d767, adobe:docid:photoshop:00d73ad4-e7ff-11de-8b71-e66145b7d767, adobe:docid:photoshop:e155aea0-e7fe-11de-8b71-e66145b7d767, adobe:docid:photoshop:e155aea8-e7fe-11de-8b71-e66145b7d767, adobe:docid:photoshop:0803244b-e7ff-11de-8b71-e66145b7d767, adobe:docid:photoshop:17bad6cc-e7ff-11de-8b71-e66145b7d767, adobe:docid:photoshop:d022d15e-e7fe-11de-8b71-e66145b7d767, uuid:2E356CBE33EADE11A8DAF3FDEF3F6791, uuid:30356CBE33EADE11A8DAF3FDEF3F6791, uuid:1E50EF8D0D8011DF8F2AC471904A1ED9, uuid:C5EB398BFCE7DE11A31D91CFBD4AF261, uuid:1E50EF8F0D8011DF8F2AC471904A1ED9, uuid:BFEB398BFCE7DE11A31D91CFBD4AF261, uuid:291BC4BDE7E7DE11A31D91CFBD4AF261, uuid:2C1BC4BDE7E7DE11A31D91CFBD4AF261, uuid:2E1BC4BDE7E7DE11A31D91CFBD4AF261, uuid:ABA3C8C2CBE7DE11A31D91CFBD4AF261, uuid:ADA3C8C2CBE7DE11A31D91CFBD4AF261, adobe:docid:photoshop:136b5f5c-e7d2-11de-8b71-e66145b7d767, adobe:docid:photoshop:2f38a3c3-e7d2-11de-8b71-e66145b7d767, uuid:AFA3C8C2CBE7DE11A31D91CFBD4AF261, uuid:BA035D2BD3E7DE11A31D91CFBD4AF261, uuid:BF035D2BD3E7DE11A31D91CFBD4AF261, uuid:C2035D2BD3E7DE11A31D91CFBD4AF261, uuid:243D6D36D7E7DE11A31D91CFBD4AF261, adobe:docid:photoshop:66e84c64-e7d5-11de-8b71-e66145b7d767, adobe:docid:photoshop:66e84c6c-e7d5-11de-8b71-e66145b7d767, adobe:docid:photoshop:55ecfab1-e7d5-11de-8b71-e66145b7d767, adobe:docid:photoshop:da98d559-e7d4-11de-8b71-e66145b7d767, adobe:docid:photoshop:4550d86a-e7d5-11de-8b71-e66145b7d767, adobe:docid:photoshop:4550d862-e7d5-11de-8b71-e66145b7d767, uuid:C7B549D80D8211DF8F2AC471904A1ED9
Web Statement                   : http://www.taylorandfrancis.com/
Marked                          : True
Format                          : application/pdf
Rights                          : © 2009 by Taylor & Francis Group, LLC
Creator                         : 
Page Layout                     : SinglePage
Page Count                      : 510

EXIF Metadata provided by EXIF.tools

A Beginner's Guide To Structural Equation Ing Beginners 3rd Ed

Navigation menu

Versions of this User Manual:

Views

Navigation